It has not been an easy season for Arsène Wenger, by any measure, but it will at least have the possibility of a happy ending and, once again, he can look his critics in the eye and ask if they will join him for the FA Cup final. It will be the eighth in his 20 years at Arsenal and it is probably worth remembering, amid all the criticism that comes his way, he has won six of the previous seven. Or, putting it another way, that his personal total is the same number Sunderland, Leeds, West Ham and Leicester have managed altogether.

Wenger is also outnumbering Manchester City, who have won the competition five times, and the fact it will be Arsenal taking on Chelsea on 27 May represents a personal ordeal for Pep Guardiola given what it means for his first season at Manchester City. Guardiola’s arrival was supposed to make City credible challengers for every trophy going. In reality, this result confirms it has been a chastening experiences – eliminated at the first knockout stage in Europe, beaten by Manchester United two games into the EFL Cup and now their latest disappointment at a time when City are 11 points off the top of the Premier League table.

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Guardiola’s team cannot even be guaranteed a place in the Champions League next season, with Thursday’s fourth-against-fifth encounter against United looking increasingly fraught with danger, and his disappointment will be exacerbated by the way this semi-final unfolded. His team had led, after 62 minutes, through Sergio Agüero’s breakaway goal and the same player also scored a perfectly legitimate goal late in the first half only for a linesman to judge, mistakenly, that the ball had gone out of play in the build-up.

At 1-1, City also hit the woodwork twice, first through Yaya Touré’s volley and then Fernandinho’s header, but Arsenal also deserve immense credit for turning the game upside-down. The competitive courage of Wenger’s team has been questioned all season but not on this occasion. Nacho Monreal chose a good time to score his first goal of the season and, 11 minutes into the first period of extra-time, the ball was at Alexis Sánchez’s feet inside the penalty area. Sánchez, a player City want to sign, was quick, alert and clinical, firing in the goal that took Arsenal, the 12-times winners, to a record 20th final.

It was a pulsating semi-final but, for City, it was also laced with controversy and they left Wembley with a smouldering sense of injustice because of the incident, five minutes before half-time, when Agüero’s goal was ruled out because of a trigger-happy linesman. Leroy Sané’s cross had never strayed out of play and, though it was a matter of millimetres, the television replays confirmed it was wrong for the linesman to raise his flag. Agüero’s shot was behind the goal-line when Petr Cech scrambled it away. Raheem Sterling, who had replaced the injured David Silva, made sure from the rebound before City’s celebrations were cut short.

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That was not the only contentious incident of a spiky match and City’s complaints will also take in another moment in the first half when Agüero went down in the penalty area with Alex Oxlade-Chamberlain in close proximity. Arsenal will argue, in turn, they could have had a penalty of their own after Sánchez claimed he had been impeded by Jesús Navas. It was a difficult afternoon for the officials and Silva was still remonstrating with the referee, Craig Pawson, as he limped away midway through the first half. Silva had taken a whack from Gabriel and, without the Spaniard, City lost their early momentum.

It was not until the early part of the second half, however, that Arsenal had their first sustained period of pressure and the paradox was that was the period when Agüero opened the scoring. Arsenal’s vulnerability to quick, incisive counterattacks has undermined them all season and on this occasion it needed only one pass once Aaron Ramsey had lost the ball to Touré close to the City penalty area. Touré, such a brilliant big-game player, aimed the ball 40 yards and Agüero had enough speed to get away from Monreal. Cech was slow to react after a slightly heavy touch from Agüero and the striker punished him with his 29th goal of the season.

After that, a more streetwise team than City might have shut the game down. That, however, is not the way Guardiola’s side operate and Arsenal, after a cautious first half, had already started to play with less restraint. Close analysis of their new 3-4-2-1 system should also note it was one wing-back who created the equaliser and another who scored it. Oxlade-Chamberlain’s cross created the danger and Monreal, two years since his last Arsenal goal, fired in a low volley.

If Cech had looked slightly ponderous for Agüero’s goal, it was a crucial save from the goalkeeper to turn Touré’s shot against the post. Yet Danny Welbeck’s arrival, as a replacement for Olivier Giroud, had also given Arsenal new impetus in attack. Welbeck applied a slight touch before Sánchez’s winner, turning the ball on after Laurent Koscielny had headed Mesut Özil’s free-kick across a congested penalty area. Three times in a row a player in red and white got to the ball ahead of anyone in blue – and on the third occasion it was Sánchez, nipping in between Gaël Clichy and Vincent Kompany, to deliver the telling blow.