Wenger's position at Arsenal had come under scrutiny again after defeats to Blackburn and Bayern Munich.
The Gunners boss seemed on course for another disappointing result on Saturday when Andreas Weimann equalised Santi Cazorla's opener, but the Spanish midfielder rescued his team with a well-taken winner five minutes from time.
Wenger ranted at his critics before the 3-1 loss to Bayern in midweek and the Frenchman was once again in a defiant mood following Saturday's win over Paul Lambert's side.
When asked whether these had been "dreadful times", Wenger replied: "Of course. But you do your job and focus on the next game. That's what I did.
"I'm not 30 years in this job at the top level to be destroyed by people saying I don't know what I'm doing. (If I was) I would have stopped this job a long time ago. But we have responsibilities and have to stand up and get the team ready to fight."
The critics who doubt whether Wenger is the right man to lead the Gunners forward may have not been fully convinced by this performance.
Arsenal dominated for large parts of the game, but they passed sloppily at times and their tendency to want to walk the ball in to the net infuriated many among the home support.
It took two bits of magic from Cazorla to rescue the north London club.
The Spaniard, who now has 11 goals for Arsenal, opened the scoring with a fierce drive past Brad Guzan after he saw his first shot blocked.
Weimann grabbed a deserved equaliser on the counter in the second half, but Arsenal threw everything at their opponents and Cazorla snatched all three points with a well-taken strike after 85 minutes.
Wenger said: "Santi Cazorla played very well and delivered in a game that was very important. He's a great player.
"He has shown he's a great football player and I'm happy and grateful he delivered those two important goals today [Saturday].
"I didn't expect him to score so many goals for us. He has 11 games to go so he could get a few more.
"For me, he's a complete football player because he also has spirit. He was a bit tired in the last few minutes, but he continued to fight.
"He's a good example to follow. He's happy to come and play and practice every day, and he gives you absolutely everything on a football pitch."
The victory means Arsenal now sit just one point behind Tottenham, who play West Ham on Monday night.
Should the Gunners beat Spurs next Sunday, they will leapfrog their bitter rivals into fourth place.
Villa, meanwhile, slip in to the bottom three on goal difference following Wigan's win at Reading.
Manager Lambert was disappointed to see his team come away with nothing despite a pleasing display.
"We shouldn't have lost it," the Scot said.
"We were exceptional in parts of that game and we should have been in the lead at half-time.
"At the very least we should have come away with a point. We've played like that for a number of weeks: West Brom, Everton, Newcastle in the second half, West Ham.
"We've just got to try and get more points on the board. We've got massive games still to come, and there'll be a lot of twists and turns this season."
Lambert's side have only won once in the Barclays Premier League since December, but he is confident they will not be relegated this season.
"I have the belief in ourselves that we'll always be okay," the former Norwich manager said.
"I don't get caught up in the whole issue. If the lads keep playing like that, that's all I can ask for."