Laurent Koscielny had the best chance of the first period at the Emirates but headed against the crossbar from close range from Wilshere's cross.
Wilshere broke the deadlock four minutes after the re-start with a close range effort after Olivier Giroud had headed down skipper Thomas Vermaelen's cross from the left into his path for his first goal since November 2010.
Podolski then played a one-two with Giroud before firing a sublime left-foot volley over Geoffrey Jourdren to seal the win on 63 minutes.
Schalke beat Olympiakos in the other Group B game, meaning Arsenal qualify with the German side ahead of their final group game in Greece in a fortnight.
But the Gunners could have been behind from a quick counter attack as Anthony Mounier got clear behind the defence and goalkeeper Wojciech Szczesny, just back in the side after an injury lay-off, had to come out to the edge of the box to make an important block.
After their two goals, the hosts looked much more assured and Montpellier keeper Jourdren palmed a floated effort from Santi Cazorla over the crossbar as the Gunners looked to extend their advantage.
Giroud was then denied by his former team-mate, who pulled off a point-blank save with his legs after the Arsenal striker was put into the right side of the penalty area by Podolski.
At the other end, substitute Emanuel Herrera shot just wide with a low effort from distance late on.
The win sees Arsenal move into the last 16, sitting a point behind leaders Schalke.
Boss Arsene Wenger will now target victory in Athens next month to keep alive hopes of topping the group and securing a place among Europe's top seeds for the knockout stages.