A Matt Jarvis header from close range put the visitors ahead after 40 minutes, but Lukas Podolski fired into the bottom corner to equalise just a minute before the break in a largely frustrating first half for the Gunners.
Arsenal made no mistake after the break - Olivier Giroud showing great composure to put them in front - before Podolski got his second of the night to wrap things up as he slammed into the roof of the net.
The result saw Arsenal move back into the top four, but that could change if Everton do not lose their game in hand at home to Crystal Palace on Wednesday night.
Just three days after seeing off Wigan on penalties to reach the FA Cup final, Wenger was pleased with the character of his side having fallen behind.
"In the first half it was difficult but the vital thing was to come back just before half-time to 1-1 and in the second half we controlled it well," he told Sky Sports.
"Overall, it's a hard-fought win but a very, very important one and as well, a very pleasant (one) because after the Saturday game we had to dig deep with some players and they showed a great mental response."
Wenger also praised Giroud for the goal which put Arsenal 2-1 in front as the French striker plucked Thomas Vermaelen's floated delivery from the air before firing through the legs of Adrian.
He said: "It was a great goal, a really great goal. First touch with his left foot and he finishes well with the right foot.
"Some credit as well to Thomas Vermaelen. He came in with a great ball but there was a lot to do after that and it was a great goal.
"If the first touch in the box is not perfect then there's no chance of scoring but that first touch was perfect."
Wenger also saluted Aaron Ramsey as the midfielder came off the bench to tee-up Podolski's second with an intelligent cushioned header.
"Aaron is a very important player for us and he has that power and that drive forward. He surprises you as well with his vision in the final third and he did that with the header.
"It came to the feet where you want it to go because Podolski is a great finisher."
The Arsenal boss is now hoping for consistency in the remaining four Premier League games of the season and says he is only concerned with the results of his own team.
"It was two comparable weeks where we needed to win. We did it. Let's continue like that," he said.
"Everybody's under pressure to do well and the best way to deal with that pressure is to just focus on your own performances."