Schalke's win over Olympiakos in the other game in Group B ensured the Gunners' qualification to the next round, and Wilshere emphasised the importance of finishing top to avoid tougher fixtures.
"I think we have to step it up from tonight," Wilshere told Sky Sports. "It wasn't good enough, especially in the first half.
"We stepped it up in the second half and got the three points, so we have to go now, watch the game, work on our weaknesses and get into the last 16 and whoever we get, you've got to play the best teams in the competition to win it so it doesn't matter who you get.
"We're pleased. It was a tough game tonight. Montpellier came tough. We started off sloppy but we knew if we kept going we'd get the win.
"We haven't finished top yet, still got one more game to go at Olympiakos, and we need to finish top of the group. We've seen in past years that if you finish second you can be in trouble going into the last 16, so we need to finish top.
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"The first half was hard and we were sloppy. We were giving the ball away and we weren't ourselves. We had to come out and get an early goal. I was lucky enough to be in the right place at the right time. It was a great header from [Olivier] Giroud and I'm happy to get the three points."
The game's other goalscorer Lukas Podolski equally stressed the importance of getting a positive result in the tie in Athens after his stunning volley sealed the points, adding that there is more to come from the Gunners.
"It was a great goal," Podolski stated. "A great pass from Giroud. It was a little bit lucky, but it was an important goal, the second goal, and we are happy with the three points and now looking forward to play against Olympiakos.
"In the first half we lost a lot of balls. But I think the first goal from Jack was good for us. Montpellier played very well but we have the three points and are very happy about this.
"There must be [more to come]. We must fight and we have a tough game against Olympiakos. It's not easy there, but we want to win there."