The former Ajax defender scored in the 75th minute at White Hart Lane with a beautifully-struck right-footed shot after a Kyle Walker free-kick from the right was nodded on by the Swansea defence.
Spurs leapfrogged Everton to take fourth place, level on 29 points with third-placed Chelsea who have a game in hand after losing the Club World Cup final in Japan 1-0 to Brazil's Corinthians on Sunday.
In the day's other match, West Bromwich Albion stayed sixth after being held to a goalless draw by 11th-placed West Ham United at The Hawthorns.
City beat Newcastle United 3-1 at St James' Park.
"We had so many chances, so many opportunities to put the match to bed even earlier," said Tottenham manager Andre Villas-Boas. "We were very solid today... it would have been extremely unfair if we hadn't got the win today."
Walker had Tottenham's best chance of the goalless first half in the 40th minute when he blasted the ball at goalkeeper Gerhard Tremmel from 25 metres.
Swansea's Michu then tried an opportunistic lob from the halfway line two minutes later that dipped over the bar as Hugo Lloris scrambled back to cover.
Swansea manager Michael Laudrup, whose side have not won at White Hart Lane in 15 league appearances there and remain 10th, was critical of referee Mike Dean's delay in halting play while his man was on the ground.
"I was very angry," the Dane told Sky Sports News. "If two players go up and you see there could be a head injury, then there's only one thing to do: The referee just has to blow the whistle.
"That was poor refereeing... we are talking about maybe a head injury and the referee let the play go on. I think that was a very bad decision," he said, adding that the Spaniard was recovering well.
