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 after 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."