The comeback was complete when Aaron Ramsey, who was returning from a long injury layoff, headed down for Podolski to fire home his second in the 78th minute.
After the match, Wenger praised his players for showing the spirit and character required to fight back and return to fourth spot in the table, replacing Everton, who meet Crystal Palace on Wednesday night.
"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 (FA Cup semi-final win over Wigan Athletic) we had to dig deep with some players and they showed a great mental response."
Wenger is now hoping the Gunners can continue their winning ways for their last four league fixtures and the FA Cup final next month.
"Everybody's under pressure to do well and the best way to deal with that pressure is to just focus on your own performances," he added.
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