Arsène Wenger praised Arsenal’s players for banishing doubts about their mental strength by coming from behind to beat Manchester City at Wembley and set up an FA Cup final with Chelsea.

Things looked ominous for Wenger’s men when Sergio Agüero fired City into the lead midway through the second half of the semi-final but Nacho Monreal equalised to take the match to extra time. Alexis Sánchez pounced in the 101st minute to score a winning goal and preserve Arsenal’s hopes of ending a tumultuous season with a trophy.

Alexis Sánchez gives Arsenal extra edge to sink Manchester City in semi Read more

Wenger said: “It was a big test for us, it was a mental test. Many people questioned that – can we turn up on an occasion like that? I am very proud for the players. They showed a united, determined and playful response.

“I want them to enjoy to play together and to play for the club and reach the targets that we set for them together. Football is a team sport and as manager your biggest joy is when you feel you have a real team performance and that was the case. That’s why I’m happy.”

Several of Arsenal’s players said afterwards that they had been motivated, in particular, by a desire to show their faith in Wenger, who in recent months has been the subject of a sustained campaign by a large section of fans who want him to be replaced as manager of the club he has led for nearly 21 years. Despite his happiness Wenger was still not minded to say whether he would still be in charge next season.

“I feel the club is in a very strong shape,” he said. “We have a very strong overall situation and a very strong team. We have shown that. That for me is the most important. One day I will leave anyway so the most important [is that] Arsenal will always be a great club that everyone admires.

“When you see the stands and see how big this club is ... when fans go home happy I feel happy. When we don’t win games I feel sad because I don’t feel I give what people want. We have gone through a very difficult period, we faced some adversity that made the situation more difficult from inside as well. We showed a united response, not a divided response. Mentally we were in a fragile position.”

While Wenger basked in the glow of an unexpected victory, Pep Guardiola was left to confront the reasons for significant underachievement. He was hired to guide City to new heights and given great riches to help lead the way do this but this defeat, on the back of premature elimination from the Champions League and floundering in the Premier League, means the club will end the season empty-handed. “We will improve next season, we will be strong,” said the Catalan, who has never experienced a season without silverware as a manager.

“Always I did in my life and always I will do in the rest of my life to be better and that is all,” he added. “That is what I want to do.”

Guardiola said the manner of the defeat by Arsenal told the story of City’s campaign this season, with critical details undermining otherwise encouraging performances. City fluffed some good chances, hit the woodwork twice and Agüero had a goal disallowed in the first half when an assistant referee deemed – wrongly according to television replays – that a cross from Leroy Sané had strayed out of play before reaching the striker.

“We arrived here and created more chances than the opponent, hit the post twice and the finishing [was bad] – it was a bit like what happened [in other matches] this season,” he said. “The team does not need to do a lot of things, we have some problems to score but we played like we want and that’s all, so in the end we are sad today, but tomorrow we have to stand up and finish the games we have still to play. We have to look inside ourselves and see the reason why we can improve in the future.”