Bale firstly set up Jan Vertonghen for Spurs' opener with a lovely lofted pass in the seventh minute, before lashing home his 27th goal of the season for club and country on 21 minutes.
Michu pulled one back for Swansea, but Bale's brilliant block on Nathan Dyer in the final minutes ensured their fightback fell short and Tottenham climbed to third in the Barclays Premier League table to boost their Champions League aspirations.
Villas-Boas was keen to praise his team after they bounced back from consecutive league defeats, but acknowledged another outstanding display from Bale.
He said: "When there are so many things that go well for the team it is hard to single out individuals but Gareth is a world class player offensively and defensively, as he showed when he made the block at the end to prevent Swansea getting a draw.
"But the whole team deserve credit because coming here after two league defeats and playing the way we did is never easy.
"We spoke at half-time about trying to get the third goal to put it beyond doubt but we couldn't and they did well to come back.
"We showed great determination and the winning mentality that surrounds us to come away and hold on for a great win from home and put ourselves back on track."
Swansea manager Michael Laudrup, one of Europe's finest midfielders during his own playing career, also praised Bale's exceptional qualities as the Welsh club suffered a third straight defeat since reaching the 40-point mark.
"They had a very good start but both goals showed a player of Gareth Bale's quality can make the difference," he said.
"With the first we could have done better even if it was a good pass and finish.
"But the second goal was just a fantastic finish by a very good player."
He added: "There are different levels of players, even at this level.
"There are good players, very good players and exceptional players and they are the ones who sometimes out of nothing can make a difference.
"If you see his goal today, or his goal against West Ham, it was not even a chance. He is a player who makes the difference and you are lucky if you have that. Not many teams do.
"If you take Bale out, without him the teams were at the same level."
Despite the loss the Dane was pleased with his side's comeback effort.
"After the two goals we reacted well and we should have scored one (through Nathan Dyer) that was harder to miss than score," he added.
"We talked about it at half-time, we felt they could be tired after having so many players on international duty and in the second half we dominated, we scored one and got close, but not close enough, to a second."