The Gunners had been at their free-flowing best in thumping Reading 5-2 on Monday but had to show a different side to their game in the 1-0 win at the DW Stadium.
The Latics more than matched their opponents on a sodden pitch but Arsenal made it three Barclays Premier League wins in a row thanks to Mikel Arteta's hotly-disputed penalty.
Arsene Wenger's side have been criticised in the past for slipping up in such games, and 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."