Wilshere back on target as Arsenal qualify
Wilshere's sharp finish shortly after half-time and the growing signs of a partnership developing between Podolski and former Montpellier favourite Olivier Giroud made it a satisfying evening for manager Arsene Wenger who has now led his side into the last 16 for the 13th consecutive season.
Podolski wasted two great chances in a frustrating first half as Arsenal looked to bounce back from a 2-0 defeat by Schalke in their previous home game in the competition, but the hosts turned on the style after the break.
Midfielder Wilshere broke the deadlock at Emirates Stadium after 49 minutes with a close-range finish, his first goal since returning from 14 months on the sidelines with an ankle injury.
Podolski scored with a superb volley just past the hour and Arsenal moved to 10 points, one behind Group B leaders Schalke whose 1-0 defeat of Olympiakos ensured that the London club qualified for the knockout rounds.
"We needed two good results, our result and the Schalke result, and we got them so we are happy," Wenger told reporters.
"The first half we had problems but in the second half we moved the ball better and scored two good goals.
"Jack had a difficult start in the game but he got stronger and stronger after the goal."
Disappointing French champions Montpellier offered only a sporadic threat as their slim hopes of a consolation place in the Europa League vanished.
If Montpellier, who are bottom of the group with one point, avoid defeat against Schalke in their final match on December 4, Arsenal could still snatch top spot with a win at Olympiakos.
"We will take a full-strength side there," Wenger said of the clash in Athens. "We will play to win the group."
Buoyed up by a morale-boosting 5-2 win over Tottenham Hotspur in Saturday's North London derby, Arsenal began brightly and nearly went ahead after 10 minutes when Laurent Koscielny thumped a powerful header against the crossbar.
It took the home side 20 minutes to threaten again when Montpellier failed to clear a ball in their area and Podolski weaved across the area before sliding a shot just wide of Geoffrey Jourdren's right-hand post.
Podolski went close again a few minutes later when his left-foot effort flew agonisingly wide of the other post after Santi Cazorla's clever pass released him.
Montpellier, who had needed to win their final two matches to have any chance of finishing third in the group, were well organised but lacked attacking threat.
A couple of breakaways and a few long-range shots were all they had to show for a tepid first half although they did manage to frustrate a disjointed side who were seeking a first home win against a French club in the Champions League.
Giroud, who has recently shown signs of the goal-scoring prowess that persuaded Arsenal to sign him from Montpellier in the off-season, helped to unlock his old club after the break.
His header from Thomas Vermaelen's cross in the 49th minute allowed Wilshere to open the scoring, the England midfielder stretching out his leg to poke a shot past Jourdren.
Wilshere celebrated by running to embrace Arsenal physiotherapist Declan Lynch.
Just past the hour Giroud linked up with the increasingly influential Wilshere and played a one-two in a congested area with Podolski, teeing up the German perfectly to lash a textbook volley past a flailing Jourdren.
"I didn't think he had one of his best games tonight," Wenger said of Giroud. "But when you measure it afterwards he had two assists tonight and defensively he is always very reliable.
"When he uses his link play he is fantastic, he can be the complete striker."
Arsenal went close to extending their lead with Cazorla forcing a fingertip save from Jourdren and Giroud was also denied late on by the keeper.