The Gunners arrived at the DW Stadium fresh from a 5-2 thumping of Reading but were pushed all the way by Roberto Martinez's men and needed a second-half Mikel Arteta penalty to secure the spoils.
Wilshere said: "We got the points and we showed a different side to our game.
"It's a tough place to come to, Wigan, and they play a strange formation compared to other teams in England. They caused us some problems but we dealt well with it.
"It was a big adjustment with the pitch and the conditions but we showed great character. We know we can come to places like this and dig out results and we have to keep doing that.
"All the big teams when they come here, maybe they don't play very well but they seem to get results and we've done that."
It capped an excellent week for Arsenal with two victories and five players signing new long-term contracts, including Wilshere.
The 20-year-old has been in very good form since returning to action in October following a year absence with an ankle injury.
"It was nice to sign the new contract and I'm thankful for all the support the club have given me," he added.
"This is a great club, everyone can see that, and for me it's like home. My friends and family are down the road, I've grown up at this club so I really
feel like part of the family.
"I think I've shown commitment to the club and they've shown it to me so I think we have a good thing going on."