The Premier League side, who are looking to end a nine-year trophy drought, fell behind to a Jordi Gomez spot kick in the 63rd minute before Per Mertesacker – whose foul on Callum McManaman gave Wigan the opportunity to lead – levelled matters with eight minutes of normal time remaining.
Alex Oxlade-Chamberlain rattled the crossbar before Arsenal were forced to confront a shootout.
They passed the test with flying colours as Lukasz Fabianski saved from Gary Caldwell and Jack Collison, while Mikel Arteta, Kim Kallstrom, Olivier Giroud and Santi Cazorla netted to seal a 4-2 triumph from 12 yards.
"We were under a lot of pressure to win and we were 1-0 down against a good Wigan side who are strong physically, strong on the ball," Wenger told BBC Radio Five Live.
"We needed to be mentally strong and we needed to be resilient. We did it and we were calm on the penalties. That was, of course, fantastic."
Wenger conceded he did not share the demeanour of his players while watching events unfold from the dugout.
"My own emotions were going through the roof," he said ruefully.
"Unfortunately that takes a few days of your life maybe, but when it finishes well you don't mind so much."
Arsenal headed into the contest winless in their past four Premier League outings, the last of those a 3-0 defeat at Everton that handed the initiative in the race for the final UEFA Champions League qualification spot to the Merseyside club.
With five league games left to remedy that situation ahead of the May 17 cup final, Wenger believes his team's narrow Wembley success can act as a catalyst.
"It will rebuild the confidence, that was very jaded, and it will give us a target until the end of the season," he said.
"It allows us as well to focus in a more serene way on the Premier League and I am confident we will respond well and come back."
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