The Belgium defender fired his first league goal for Spurs following his £12million move from Ajax when he showed composure to sweep a free-kick from Kyle Walker past Gerhard Tremmel.
It was the reward Tottenham deserved after controlling a disappointing encounter at White Hart Lane that failed to match expectations before exploding into acrimony deep into injury time.
Swansea striker Michu was caught by Hugo Lloris as the goalkeeper punched the ball clear and fell heavily to the floor.
While those players closest to the stricken Michu furiously waved to the touchline for medical assistance, Spurs played on through substitute Andros Townsend.
Incensed by Townsend's perceived lack of sportsmanship, defender Chico Flores confronted the forward and several Spurs players and was only placated when his Spanish compatriot Michu climbed to his feet.
Sorely missing the penetration of the injured Gareth Bale, Spurs struggled to break down opposition that defended stoutly without offering much threat themselves.
Andre Villas-Boas could be heartened by his side's third clean sheet of the season and the energy of the irrepressible Jermain Defoe, but playmakers Clint Dempsey and Mousa Dembele offered little.
Emmanuel Adebayor limped off with a second-half injury and there was a brief injury-time cameo from Scott Parker, making his first appearance since suffering an Achilles injury at Euro 2012.
Spearheaded by the Premier League's top scorer Michu, Swansea rarely troubled the home goal and looked lethargic after their midweek Capital One Cup quarter-final victory over Middlesbrough.