The Gunners offered the perfect response to recent FA Cup and UEFA Champions League setbacks as they burst out of the blocks at Emirates Stadium, with Santi Cazorla firing them in front with just six minutes on the clock.
They were unable to build on that advantage, though, and saw Andreas Weimann level in the second half.
It appeared as though the spoils would be shared heading into the final five minutes, but the Gunners' relentless pressure finally told as Cazorla struck again to leave the visitors empty-handed.
"I think that was the main thing, it was relief," Wilshere told Sky Sports.
"We dominated the whole game and I think if we drew that game it would have been criminal. So it was a massive relief and thankfully we got the three points today.
"Maybe when you are 1-0 up and you aren't comfortable, you need that second goal for a bit of protection. We didn't get it and Villa kept pressing and pressing and beat us on the counter.
"They got a goal on the counter, which is never good for us. We had a corner and then they scored, so we will have to have a look at that.
"We showed great character in the end and now we are on a good run in the league so we have to keep that going, we have to forget about the disappointments we had in midweek and last weekend and push on in the league."
Victory over Villa, their third in a row in the league, has lifted Arsenal to within a point of the top four and Wilshere is looking for Arsene Wenger's men to carry their momentum into next weekend's derby date with arch-rivals Tottenham.
He added: "It was a massive win today. Everton lost as well, so the top five are starting to pull away now and we have to keep this run going.
"We are on a good run in the league and have got a big game next weekend against Tottenham - big for us, big for the fans - and they are the games you want to be involved in as a football player.
"We know now what we have to do and we have to step up."