Spurs trailed 1-0 for a large part of the game after Samir Nasri's opener in the fifth minute, but they drew level thanks to Clint Dempsey in the 75th minute and further goals from Jermain Defoe and Gareth Bale secured a vital win in the race for the top four.
Defoe was one of three players to make an impact off the bench, with Lewis Holtby and Tom Huddlestone both providing assists, but Villas-Boas played down his own tactical decisions after the final whistle.
"I must praise the team for their fighting spirit and it was great to see them turn it around in the second half because the first half was not to our level," Villas-Boas told Sky Sports.
"We improved dramatically and the players were fantastic to get this result.
"It comes down to the character of the players. All three that came on showed that desire and ambition to help the team, and in the end we were able to punish them."
The result kept Tottenham in fifth but they are now within two points of Arsenal, with a game in hand, in a three-way battle that also includes Chelsea.
Villas-Boas knows there are still a number of key games remaining this season but was delighted to beat City and keep the pressure on.
He said: "It is very important. We can't say how significant it will be until the end of the season.
"It is very competitive and will probably go down to the wire.
"It was important for us to win today. Arsenal play Manchester United next week and that is an important fixture.
"We have to continue getting as many points as possible and see if the decisive fixture is that game against Chelsea."