City has moved to within seven points of City following the win.
"Our objective was to shorten the distance to the leaders today, and we managed to do exactly that, but we don't take extra flavour from being the first team to beat them," Villas-Boas told Sky Sports.
"We needed to do our job to get our Premiership challenge back on track, and I think it's pretty much alive now.
"I think the fact that also other teams won, Man Utd won, Arsenal won all of us are chasing now and all of them are putting pressure on Man City so a lot of team still competing in this league.
"Again December fixtures will be decisive, Tottenham still with a game in hand so it's very hard at the moment."
"I always thought it was going to be a game that could be decided by small detail," added Villas-Boas.
"So we had to keep calm even with ten men I think Man City could become even more dangerous we had to keep out patience.
"I think the players did brilliantly because I think today they adapted to the circumstances and felt the game and knew Man City with ten men and negative result they are always a threat.
"We were very patient to try to get the goal we deserved to take us the win.
"It was a very competitive game, tremendous from both teams and I am very, very happy for my players they deserve this win."