The club's 2-1 success over Manchester City booked a Cup final date with Premier League leaders Chelsea on May 27.
Wenger's side were on the brink of yet another painful defeat in a turbulent season after being out-played by City and falling behind to Sergio Aguero's second half strike at Wembley on Sunday.
It was exactly the kind of do or die situation that has been the downfall of Wenger's notoriously fragile team in the past.
But for once Arsenal rose to the challenge, showing unexpected levels of grit and determination as they drew level through Nacho Monreal before Alexis Sanchez grabbed the winner in extra-time.
The 2-1 success booked a Cup final date with Premier League leaders Chelsea on May 27 in what will be a record 20th appearance in the showpiece match for the Gunners.
And for Wenger, it was an encouraging sign that Arsenal have enough mental strength to ride out the storm currently buffeting his club.
"I felt it was a big test for us, a mental test because many people question if we can turn up on an occasion like this," Wenger said.
"That's why I'm very proud of the players because they stayed united and determined. We gave a strong response.
"We had a little bit of a cautious start, but we didn't give chances away and we grew into the game.
"I want the players to enjoy playing together and to play for the club. That is the most important thing and to reach the targets we set together.
"Football is a team sport and as a manager the biggest joy is when you feel like you have a real team performance."
In the short term at least, Arsenal's progress to the final might buy Wenger some welcome breathing space amid constant calls for his resignation from angry fans frustrated at their team's struggles in recent years.
Wenger hasn't signed an extension to the contract that runs out at the end of the season and is yet to reveal whether he will remain in charge next term.
The 67-year-old is well aware that failure to lift back into the Premier League's top four, which brings qualification from the Champions League, over the final weeks of the season will trigger a fresh barrage of abuse from the doubters.
But he is reportedly keen to stay and, after once again dodging the chance to make his future plans clear, he emphasised Arsenal still have a bright future despite their recent woes.
"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, we have shown that today and for me, that is the most important," 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."
Sunday's success was also a small vindication of Wenger's decision to switch to a back three for Arsenal's last two matches, both of which they have won.
"What I did is just to give a bit more security to a team who conceded three goals in the last three away games," he said.
"At some stage you need to focus on something for the players which creates some confidence.
"I had it in mind for the Manchester City game after we won at Middlesbrough and got some confidence back.
"No matter what you do, even if you get it right at some stage, people say 'why didn't you do it earlier?"