The Gunners initially fell behind in the tie as Jordi Gomez fired home a penalty and the tie looked bleak for Arsenal before Per Mertesacker's late equaliser forced extra time.
With no winner found during 30 minutes of extra time the game went to penalities where misses from Gary Caldwell and Jack Collison proved costly for the Latics.
Speaking about the game Wenger said: "[I am] relieved because we were under big pressure because I expected a difficult game and I was not disappointed.
"Wigan were impressive, well-organised, physically very strong, calm when they had the ball and they played very well. They have shown why they knocked Man City out in the quarter-final and we had everything a little bit against us for a while in the game. We needed to show some character and some resilience and as well some nerves in a very united and resilient way.
"We have seen again today that it was a cup game and the difference between winning and losing is very, very small and this game has shown that again.
"But even if we had lost I would have said that we were at the mental level that I expected, and hopefully that win today will give us a lift for the end of the season.
"It's important mentally that we didn't go out tonight. If you imagine the consequences of going out tonight, it's quite worrying.
"I believe we have a good team, we have many players out and this morning Oxlade-Chamberlain was still not sure to be available, Gibbs was not available, plus all the other players who are out. We have a good squad but recently we were missing too many players."
Arsenal will discover who they play in the final today as Hull City and Sheffield United battle it out in the second semi final.