Arsenal won 2-1 thanks to a late Santi Cazorla strike, to bounce back from FA Cup and Champions League losses to Blackburn and Bayern Munich.
"The result was vital today," Wenger said. "We absolutely had to win this game. We couldn't afford to drop points for two reasons: we're in a catch-up situation in the Premier League, and we've come out of two massive disappointments.
"But we were focused, united, a bit nervous … we played a bit with the handbrake on in some situations and the second goal didn't come until the end.
"But we were also strong because, when Villa came back to 1-1, we found the resources to win the game. I'm proud of the attitude of the team. We'd lost against Blackburn and Bayern here, at home, and when people come to the Emirates Stadium they should not be going away with wins. Personally, you do your job and focus."