After a bright start by Arsenal, it was Wigan who went ahead through a penalty from Jordi Gomez just after the hour.
However the Gunners made it 1-1 with seven minutes left through Per Mertesacker, whose foul had given away the spot-kick.
"It is important that mentally we didn't go out. You could imagine the consequences of going out, it is quite worrying," Wenger said.
"I am relieved because there was a big pressure on us.
"I expected a difficult game and I was not disappointed.
"Wigan were impressive, well organised, physically very strong, calm when they had the ball.
"They played very well and showed why they knocked Man City out in the quarter-finals.
"We had everything a little bit against us for a while in the game. We needed to show some character, resilience and some nerves and we did it in a very united and resilient way.
"That was the most pleasing aspect because everyone expected us on that front.
"Even if we had lost, I would have said we had the mental level that I expected, and hopefully that win today will give us a lift for the end of the season."
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