Wenger's position at Arsenal had come under scrutiny again after defeats to Blackburn and Bayern Munich.
He seemed on course for another disappointing result 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 the 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."
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. 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."
- Sports & Recreation
- Santi Cazorla