Stand-in goalkeeper Lukasz Fabianski was the hero as his two saves helped Arsenal beat the holders 4-2 to reach their first FA Cup final since 2005.
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.
After Alex Oxlade-Chamberlain had hit the crossbar in extra-time, it was down to penalties.
The relief was clear for all to see as Wenger punched the air, and the Arsenal players rolled around in a joyous heap on the Wembley turf, their 50,000 fans in delirium.
Earlier in the afternoon, Everton's win over Sunderland had pushed Arsenal out of the top four of the Barclays Premier League, and Wenger - who has yet to agree a contract extension past the end of the season - was all too aware of the impact of another morale-sapping defeat.
"It is important that mentally we didn't go out. You could imagine the consequences of going out, it is quite worrying," the Frenchman 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."
Wenger hailed the professionalism of Fabianski, who is set to leave the Emirates Stadium in the summer when his contract expires.
"Fabianski did very well. He is a good goalkeeper, he showed that again today," the Arsenal manager said.
"During the game, even with the penalty they scored, he went on the right side. He is quick. In the penalties he put us in a good position."
Wenger remains confident Arsenal can now go on to finish the campaign in style.
"If we win our games, we will finish in front (of Everton), so for us it is important now to focus on the Premier League," he said.
"If we do that well I am confident we will come back, and on top of that we have a target now for the end of the season - the FA Cup final.
"We have now a period where we can concentrate on the Premier League."
Wigan manager Uwe Rosler felt his team could not have given any more.
He said: "We faced Arsenal, one of the top teams in the country and did very well, we were only eight minutes away from winning, but we had to make some substitutions early and unfortunately we could not hold out.
"We forced Arsenal to play some long balls in the second half, which is unusual for the way they play, and we still looked to attack them when we could.
"I am absolutely proud of how they performed and what they put in.
"When you can see the finish line, but just not get over and then bounce back in extra-time, that shows mental strength and character.
"When you are then going into penalties, there is always a bit of a lottery."
Rosler added: "I got asked 'have you practised penalties?' I said, 'no, I am German, you know we don't need to practice penalties'.
"We want to be back here in May (for the play-offs), when we are back in May, then we will practice penalties."
The Wigan boss had no issues with his side's attitude in the shoot-out.
"There is no criticism whatsoever on Gary, I felt we selected the guys who were freshest and had confidence.
"Gary is a leader, a man and did himself justice after having not played for so many months.
"He was ready, I thought he would score. Unfortunately he didn't. That is life, but you have to get on with it.
"Gary will recover, we will recover and move on.
"You need a bit of luck and we did not have it on our side today."
Wigan, relegated from the top flight under former boss Roberto Martinez despite their FA Cup final success, remain very much in the hunt for promotion.
Rosler said: "I guess a lot of people will say 'now they will never recover mentally from this', but we will just use t his now - last season at Brentford, we lost on automatic promotion in the 95th minute (of the final day) and the team recovered to go to the Wembley final.
"It is exactly that mental strength, that willpower and desire to get through the play-offs, that is definitely our target, that is what we are breathing, eating and sleeping, and I think that will happen."
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- Lukasz Fabianski
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