Goals from the in-form Andros Townsend and Roberto Soldado secured the points for Spurs, who had lost 3-0 at home to West Ham United before the international break.
Villas-Boas was pleased with the response from his players, with the result leaving Spurs in fifth position and three points behind Premier League leaders Arsenal.
"It was difficult in the first minutes," Villas-Boas told Sky Sports. "It was very unpredictable because Villa kept putting some very long balls in behind us, so they kept stretching us.
"We had to put our act together and I think we did after the 20th minute. We got hold of the ball and started creating things.
"In the second half, we were really, really good. After we reached 2-0, we could have gone on to score more.
"After the setback, particularly at home, we have to respond and I think we did it in a good fashion to keep ourselves in touch with the top.
"I think we deserved the result. We want to build on from here and continue to gather the most amount of points as possible."
The Spurs boss also praised the role of Townsend, who came into the game on the back of an impressive brace of performances for England and signing a new contract.
"I think he's done what he has been doing excellently," said Villas-Boas of Townsend.
"He takes the team forward, and he has the ability to drive and jump the man whenever he has the ball.
"He is always a threat and teams have started to be more aware of his strengths. He started not really well, but improved during the game."
Soldado secured the win with his first Premier League goal in open play and Villas-Boas was pleased with the display of his summer signing from Valencia.
He added: "It was a great performance. He was making the most out of his opportunities and this is the reason why we bought him."