Sergio Aguero had put City ahead with a fine breakaway goal just after the hour having had another strike controversially ruled out -- but Nacho Monreal equalised.
City twice hit the woodwork in the latter stages of normal time through Yaya Toure and Fernandinho but Sanchez, recently linked with City, pounced for a 101st-minute winner after a goalmouth scramble.
Arsenal's victory vindicated under-pressure Arsene Wenger's decision to play with three at the back for the second time in a week - but they were outplayed for large spells early on.
City's pre-game suggestions Gabriel Jesus could be fit proved a smokescreen but the Gunners struggled to hold them back and were forced to employ crude tactics.
David Silva was an early casualty, forced off with 23 minutes gone after failing to recover from a heavy Gabriel tackle. Aguero needed treatment after being kicked in the back by Laurent Koscielny and Sanchez was booked for a challenge on Fernandinho.
Before he left the field, Silva forced a good save from Petr Cech with a header and his replacement Raheem Sterling volleyed wide.
Aguero wanted a penalty after tangling with Alex Oxlade-Chamberlain but nothing was given.
Arsenal offered little but Claudio Bravo, preferred to Willy Caballero, did get some sarcastic cheers when he safely held an Olivier Giroud header.
Koscielny had a goal correctly ruled out for offside but controversy erupted when City had a goal disallowed in more contentious circumstances.
Aguero thought he had scored with a shot that crossed the line before Sterling smashed home a rebound off Cech - but the ball was ruled to have drifted out of play beforehand from Leroy Sane's cross. Replays suggested it was a very tight call but ultimately wrong.
Arsenal responded well to the reprieve with a positive start to the second half but City had another good opportunity when Nicolas Otamendi headed wide.
The opening goal came just after the hour as City broke from deep following an Arsenal corner.
The ball was headed out as far as Yaya Toure and he released Aguero with a superb pass from just outside his own area. The Argentinian raced clear of Monreal, closed in on goal and beat Cech with a brilliant flick. It was his 30th goal of the season, his 12th in 12 games and sparked jubilant scenes in the City end, including a 'Poznan' celebration.
Their joy was short-lived, however, as the impressive Oxlade-Chamberlain whipped a dangerous, dipping cross into the City box and Monreal volleyed in at the far post.
Again questions were also likely to be asked of Bravo, who made no attempt to come for the cross, but City's response was strong.
They hit the woodwork twice in quick succession with Cech tipping a hooked Toure shot onto the post and Fernandinho thumping a header against the bar.
But Arsenal went close to a late winner when substitute Danny Welbeck drove wide and it was they who made the better start in extra time.
Rob Holding headed narrowly over and moments after City surprisingly, but perhaps because of injury, took Aguero off the Londoners snatched what proved the winner. City failed to clear a free-kick into the box and although Welbeck miskicked, Sanchez made no mistake as he pounced from close range.
City went close to an equaliser when Fabian Delph had an effort deflected into the side-netting and Toure also curled a shot wide.
|Arsène Wenger reacts to the final whistle at Wembley and confirmation of Arsenal’s place in the FA Cup final. Photograph: Toby Melville/Reuters|
Arsène Wenger and Arsenal enjoy rare day of happiness in fraught times
Just for a moment, unfettered by the constricting context of this turbulent season and the grisly dance around the topic of that contract, the sun broke through. Arsène Wenger stood in his technical area and the body language all around him was positive. Gabriel Paulista – a player apparently reborn during the course of the match – won another battle of wills with the threatening Leroy Sané. Mesut Özil and Alex Oxlade-Chamberlain shared a boosting handshake as they passed each other on the pitch. The German juggled the ball casually as he waited to restart the game. Behind them, the Arsenal supporters waved their flags and basked in the glow of the goal from Alexis Sánchez that eventually put their team into yet another FA Cup final.
In that pause in the match nothing much mattered apart from the fact that for once just about everyone associated with Arsenal looked happy. The possibility, the hope, that all football clubs need at a very basic level, opened up. There are still big conversations to be had but that’s for another day.
In the past couple of years the FA Cup has been Wenger’s saving grace. The Wembley experience has not always been enjoyable – the equivalent semi-final when his contract was last winding down was one of the most excruciating matches of his 21-year tenure. Arsenal squeaked past Wigan on that occasion in 2014 and Wenger spent most of the afternoon pacing, barely able to watch, wearing a tortured expression before release came via a penalty shootout. Football, bloody football.
The strangest thing happened to Arsenal during this semi-final. Over the course of the 120 minutes they visibly rediscovered how to battle and scrap, to reboot the confidence that had shrivelled up during a generally fraught 2017.
Confidence, as Wenger has often said, is the easiest thing to lose and the hardest thing to gain. For a team whose belief has resembled melting marshmallows during their worst humiliations this season, this tussle against Manchester City allowed Arsenal to reacquaint themselves with that invaluable friend. Wenger’s patented handbrake, which has been sticky for ages, slowly loosened up.
Even when he looks half-finished the old fighter in Wenger never feels like giving up, and the tweak in his tactical strategy turned out to be an important factor. A largely fruitless search for a reliable lineup since the turn of the year – one with the required mix of attacking composure and defensive stubbornness – led him to break with tradition and shuffle his pack tactically. The back three suddenly introduced at Middlesbrough last Monday (last used 20 years before) was retained at Wembley against far more dangerous opposition. It was a mechanism designed to try to rebuild faith, to try to remind the players how to play football without tension, as much as anything else.
Initially there were some familiar sights that a tinkered tactic could not fix. A propensity to stand off, allowing City periods in the opening exchanges to have a quick think about life, the universe and everything before picking a pass, was alarming.
Riding their luck in the first half, with City’s frustrations about the loss of the midfielder David Silva to a stray knee in the back of a thigh compounded by a disallowed goal, one thing Arsenal did show was an improved sense of how to scrap and hang on in there.
On the back of that they began to feel some freedom, some enthusiasm, returning to their game. Wouldn’t you believe it, that suckered them into conceding the opening goal.
With so many Arsenal players up the pitch in search of a breakthrough, Aaron Ramsey’s concession of possession was calamitous. A pinpoint release from Yaya Touré sparked one of those moments when every pair of eyes in the stadium widens, when every heartbeat stutters for a moment to see what happens next. As it happened Sergio Agüero’s touch was heavy enough to have given Petr Cech the encouragement to consider running out to claim but the experienced keeper hesitated. Agüero had all the time in the world to execute his finish.
It felt ominously telling. The three centre-backs selected by Wenger were all high up the field when Ramsey blundered. Switched formation, same old Arsenal? Not quite. Where a situation like this has led to panic in Wenger’s ranks recently, this time they refocused quickly and energetically. The equaliser was another consequence of the new tactics as the move was made by a combination between the two wing-backs. The excellent Alex Oxlade-Chamberlain, Arsenal’s most direct and inventive presence in recent weeks, hooked the ball across for Nacho Monreal to jab in at the near post. With his wrong foot. This was different, all right.
Arsenal’s resolve, in responding rather than folding at the hint of a setback, stood them in good stead. Although they rode their luck when City twice struck the frame of the goal, as the game wore on they showed more intensity at both ends of the pitch than is normally the case when they are exposed to big game pressure.
Arsenal, and Wenger, march on.
Arsene Wenger: Arsenal answered their critics in FA Cup win over Manchester City
Arsene Wenger believes Arsenal answered their critics with a "strong and united" performance as they beat Manchester City to reach a third FA Cup final in four years.
Alexis Sanchez scored an extra-time winner as the Gunners came from behind to win 2-1 and reach a record 20th final - and an eighth under Wenger.
"People questioned us, we went through tough times," Wenger said.
"You can be divided or united and we have shown the right response."
Wenger, 67, has come under more scrutiny this season than at any other time in his 21-year reign at Arsenal, with the Gunners lying in seventh place in the Premier League and on the receiving end of a 10-2 aggregate thrashing by Bayern Munich in the Champions League.
The Frenchman is out of contract at the end of the season and has been offered a new two-year deal, although he is yet to announce whether he will continue.
Some sections of Arsenal fans have protested against Wenger in recent months, but the manager was pleased to see his side rally after falling behind to Sergio Aguero's opener.
"You know I feel the club is in a very strong shape, and that we have a very strong overall situation and a very strong team," he said.
"One day I will leave anyway so the most important thing is that Arsenal will always be a great club that everybody admires.
"I felt it was a big test for us today, a mental test because many people question if we can turn up on an occasion like this.
"It was a very tight game but overall I think we deserved to win the game. The players showed great togetherness."
'Sanchez will be here next year'
Sanchez and Mesut Ozil have been linked with moves away from the club this summer, and the Chilean highlighted his importance with his 24th goal of the season which settled the tie,
Wenger expects the former Barcelona forward to extend his stay at the Emirates.
He said: "Alexis Sanchez was like the team. He had problems at the start and became stronger and stronger.
"He is an animal, always ready to kill the opponent. He will never give up.
"He will be here next year because he has a contract and hopefully we will manage to extend him."
'We'll improve next season'
Manchester City manager Pep Guardiola said he had no regrets after seeing his side beaten, a result which means the former Barcelona and Bayern Munich boss will end a season without a trophy for the first time in his coaching career.
"We performed like we would want to in a final," he said.
"We did absolutely everything. Congratulations to Arsenal. We'll improve next season.
"We competed here, we had more chances but the finishing was like it has been throughout the season."
'A display of steel, grit and resilience'
BBC Sport's chief football writer Phil McNulty at Wembley:
Arsene Wenger's mask slipped as he pumped his fists in triumph repeatedly at the conclusion of Arsenal's win.
Wenger has never been under greater scrutiny or pressure than he has been in recent months, dealing with the toxic combination of growing unrest among Arsenal fans and a collapse in form that leaves their Champions League ambitions under threat.
And yet here, with the pressure at its highest, Wenger coaxed the sort of performance out of Arsenal that puts him on course for an historic FA Cup win when they face Chelsea in the final in late May.
Yes, his Arsenal side enjoyed large portions of good fortune with Raheem Sterling's disallowed goal and efforts against the woodwork from Yaya Toure and Fernandinho - but Wenger deserves credit for persevering with a three-man defensive system that goes against his natural instincts.
This was an Arsenal display of steel, grit and resilience topped off by a comeback crowned by Sanchez's poached winner.
And Arsenal's players played for their manager, with some dedicating the victory to the man who has been a convenient shield for their own shortcomings during the recent slide.
Reality dictates that Arsenal's ills and the uncertainly surrounding Wenger cannot be wiped away by one win. He will need a trophy and/or that top-four place otherwise they idea of him staying on and signing a new deal will be a hard sell to that disgruntled strand of Gunners support.
As for Wenger's opposite number Guardiola, he may have had a hard luck story to tell as he left Wembley but the bottom line is the man brought to Manchester City to move them on to the next level in succession to Manuel Pellegrini will end the season empty-handed and that is a serious disappointment.
This was not how it was meant to be but this City team simply has too many flaws. For all their attacking riches, they are not ruthless enough, are insecure at the back and uncertainty continues over the goalkeeping position, where Guardiola's choice to replace Joe Hart, Claudio Bravo, has not convinced.
And now the pressure is really on - if Guardiola fails to guide City into the top four then this season will be nothing other than an abject failure after the exit at the last 16 stage to Monaco in the Champions League.
City are currently fourth, a point ahead of Manchester United before Thursday's derby at Etihad Stadium.
A top-four place was the very minimum requirement for Guardiola and his players before the start of the season - failure to deliver would not be well-received by the club's fiercely ambitious Abu Dhabi-based hierarchy.