The Gunners boss 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.
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."
He added: "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.
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."
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."
Source: PA
Source: PA