The Gunners had to fight hard to earn the 2-1 win.
"We've started badly in the past few games but we started well here," Wilshere told arsenal.com. "I think we needed that. If we'd gone 1-0 down, it could have been a whole different game.
"We knew we had to start well and Santi got a great goal. We pushed on from there. They caught us on the counter but we showed great character in the end.
"When you're 1-0 up, you're not comfortable and need that second goal to give you a cushion. We didn't get it and Villa kept pressing and pressing. They hit us on the counter and got the goal. We had a corner and then they scored so we have to look at that.
"We're on quite a good run in the league. To keep that going, we have to forget about the disappointments we have had in the cup and in midweek and push on in the league now.
"We have to keep this run going. We've got a big game next weekend. It's big for us, big for the fans and as a player, those are the games you want to be involved in. We know what we have to do and we have to step up."