The Gunners came from behind to win a thrilling contest 2-1 at Wembley on Sunday, booking a final date against London rivals Chelsea next month.
Arsenal have faced fierce criticism for a campaign that has seen them crash out of the Champions League at the last-16 sage and sit seventh in the Premier League with under a month of the season to play.
But the Gunners are now one win any from winning a third FA Cup in four years and Wenger has been thrilled by his players' response to adversity.
"I am very proud of our performance and of the spirit we have shown - I believe we gave a good answer to people who questioned that [spirit]," Wenger said to BT Sport.
"Overall we became stronger as the game went on. First half we were maybe a bit cautious and accepting to be dominated, but in the second half we became stronger and stronger.
"In the end we won the game. Of course it was a very tight game but I thought we deserved to win it in the second half.
"We went through a difficult period. I will talk more about that one day. But I think the players have shown great togetherness."
20 - Arsenal have now reached more FA Cup finals than any other team in the competition's history. Magic.— OptaJoe (@OptaJoe) April 23, 2017
The Frenchman added to BBC Sport: "When at Arsenal Football Club we want the team to turn up in big games and I think we have done that.
"Overall I am very pleased. People questioned us, we went through tough times. In these times you can be divided or united and today we have shown the right response."