Arsene Wenger warned Arsenal's title rivals that his side have the fight to win the Premier League after the Gunners went back to the top of the table with
Jack Wilshere and Olivier Giroud gave Arsenal a two-goal lead at half-time but they had to dig in after Christian Benteke pulled a goal back.
"It is a big win for us," said Wenger. "The first half was about playing our game with passing and control, the second was more about fighting."
He added: "Our defenders kept us in the game. It was an anxious finish and we had to fight right to the end and we were committed to every ball."
Villa were booed off at half-time after a limp display as Arsenal looked certain to take all three points up until the final 25 minutes.
But Villa then rallied after the introduction of forward Andreas Weimann, with the visitors having to withstand heavy pressure after Benteke headed home, the Belgian's first goal since September.
Wenger, whose side sit one point above Manchester City and two in front of Chelsea, said: "We had to have a strong response because everyone was looking at us to see if we would drop points after everyone else had won over the weekend.
"You are under a bit more pressure when you play last because you know you are not allowed to drop any points.
"The only way you can give an answer to that is to win the games and we have here."
Wenger added that he believes Wilshere is now getting back to his best after playing a key role in the victory.
The England midfielder has seen his career interrupted by injuries, but he scored Arsenal's opener on 34 minutes and set up Giroud for a second at Villa Park less than a minute later.
Wenger said: "He has got his change of pace back. He is not only a passer but he gives us an incision with the ball and for that you need to be at the top of your game physically.
"When you know the players well you can see that, and in his last games he is coming back to his best."
Alex Oxlade-Chamberlain returned as a late Arsenal substitute for the first time since the opening day defeat by Villa.
He replaced Tomas Rosicky, who sustained a nose injury in an aerial challenge with Gabriel Agbonlahor, and Arsenal will check on defender Nacho Monreal on Tuesday to see if he has a broken metatarsal.