Spurs go fourth as Vertonghen sinks Swans
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.
Manchester United, who beat Sunderland 3-1 at Old Trafford on Saturday with Robin van Persie and Wayne Rooney both scoring, have 42 points and are six clear of Manchester City at the top of the table.
City beat Newcastle United 3-1 at St James' Park.
Spurs, without the injured Gareth Bale but with England midfielder Scott Parker returning as a late substitute for his first appearance of the season after Achilles surgery, enjoyed most of the possession but were frustrated by a flowing Swansea side that proved solid in defence and dangerous on the break.
"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.
Tremmel had barely time to raise his hands, with the ball ricocheting off his shoulder without Spurs being able to capitalise.
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.
The game ended in controversy after Michu was felled by Lloris when the goalkeeper tried to punch the ball clear. With the Swansea player still lying in the area, and others beckoning for assistance, Tottenham played on to consternation from the visitors.
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.