Tottenham are up to third place in the Premier League, leapfrogging Chelsea with a win that was harder to come by than they might have expected against opponents with little to play for and rumoured to be on the beach already, psychologically speaking.
The Swans have now lost three of their past four games since crowning their second season at elite level by winning the Capital One Cup at Wembley last month, but suggestions that they are no longer "up for it" were belied by a vigorous second-half fightback that would have brought them a point in stoppage time but for a last-ditch block by Gareth Bale to keep out Nathan Dyer's shot.
André Villas-Boas said beforehand that Spurs needed to win six of their last eight games to be certain of a place in next season's Champions League and the first was the product of two goals as good as any that will have been scored anywhereon Saturday.
They were scored by the outstanding players on view, who were in tandem each time. The first was superbly taken by Jan Vertonghen after an incisive through pass from Bale, who then scored his 22nd of the season from the edge of the D when Vertonghen reciprocated.
Michu, with his 20th goal of a remarkable debut season in British football after 71 minutes, condemned Villas-Boas and his team to an anxious finale, and Dyer threatened to deny them at the death, but the final whistle left Tottenham two points ahead of Chelsea, who have a game in hand.
As might have been expected, Tottenham played Swansea at the passing game that has served the Welsh club so well and this time Spurs were better at it in the first half, and were two goals to the good after 20 minutes.
They took the lead after seven when Vertonghen, who initiated the attack, then showed a forward's close control and finishing expertise, bringing down a lovely return pass from Bale before shooting coolly past Michel Vorm from a central position, 10 yards out.
"An excellent goal for a central defender," Villas-Boas said. "It was similar to a couple I saw him score for Ajax, coming out with the ball like that. He has a reputation for it in Holland."
It should have been 2-0 five minutes later, but Emmanuel Adebayor ran clear, leaving Ashley Williams treading water in his wake, only to shoot straight at Vorm.
The Swans ought to have drawn level when Michu wasted an inviting opportunity, heading wide across goal from Jonathan de Guzman's corner from the right. A wretched finish appeared all the more culpable within a minute when Bale produced a goal that was a sublime, even by the stratospheric standards the Welshman has set this season. Supplied by Vertonghen 22 yards out, he took a touch with his right foot before letting fly with his left, the siege-gun shot bulging the top right corner of Vorm's net. Of Bale's latest man-of-the-match performance, Villas-Boas said: "Offensively and defensively, he is a world-class player, but I'm reluctant to single out individuals because today the team deserve more credit than any one man."
After their slow start, Swansea waxed stronger the longer the game went on and they should have been back in contention before half-time when Dyer, at full pelt, headed Dwight Tiendalli's right-wing cross against the crossbar from three yards out. Even allowing for the winger's Lilliputian stature, it was an embarrassing, and ultimately costly, miss.
Michu was only marginally less culpable, this time from Wayne Routledge's cross, but then he put the points back into the melting pot when the Spurs defence left him ridiculously unattended to head home Ki Sung-Yueng's corner via the far post.
Swansea had the upper hand throughout the closing stages and Dyer would have equalised after 90 minutes but for Bale's lunging intervention.
Relieved, Villas-Boas hailed it as "a great win", adding: "It was an extremely important result for us against a team that played really well. At half-time I spoke about the need to score a third goal, to put the outcome beyond doubt, but we couldn't get it and they came back into the game strongly."
Michael Laudrup said: "One player made the difference. At this level there are good players, very good players and exceptional players. Exceptional players make something out of nothing. Bale's goal, like the one against West Ham, was out of nothing. It was not even a chance, and you are so lucky if you have a player like that."