Despite dominating possession, Spurs fell behind to a neat close-range finish from their former winger Wayne Routledge.
The visitors were camped in their opponents’ half for long periods but were frustrated by a combination of their own lack of a cutting edge and Swansea’s diligent defending, before Dele Alli eventually broke through to convert Christian Eriksen’s cross after 88 minutes.
Son Heung-min fired Tottenham ahead in added time, and then Eriksen completed the remarkable comeback with a curling effort.
While second-placed Spurs continue to breathe down the necks of leaders Chelsea, a third defeat in four games sees Swansea drop into the relegation zone.
Having cut Chelsea’s lead at the top of the table to seven points with Saturday’s win at Burnley, Spurs were aiming to further reduce that deficit with a fifth successive Premier League victory.
Yet for all that Mauricio Pochettino’s side had impressed this season, that triumph at Turf Moor was only their fifth away league win – and their vulnerability on the road was evident at the Liberty Stadium.
Although they enjoyed near total control of possession and territory during a one-sided first half, Tottenham fell behind after Swansea’s first attack of the game.
Jordan Ayew muscled his way into the visitors’ penalty area and pulled the ball back to Routledge, one of four former Spurs in the hosts’ line-up, and the winger squeezed his finish past ex-Swans keeper Michel Vorm.
But as they had demonstrated in their previous seven games without Kane – four wins and three draws – Spurs can cope in the striker’s absence.
They left it late, with Alli tapping into an empty net from Eriksen’s wicked low cross, before Son finished from close range to send the visiting Spurs fans into raptures.
Eriksen then added a third in added time to complete a stunning fightback and leave their opponents crestfallen.
A return of just one point from their previous three games had seen Swansea sink deeper into relegation trouble, one point and one place above the bottom three.
Head coach Paul Clement spoke of a nervousness inside the Liberty Stadium during Sunday’s goalless draw with Middlesbrough, though any creeping sense of anxiety for the home fans was eased with Routledge’s early goal.
They made their ground a cauldron of noise, roaring their approval with every tackle, block or pass from a Swansea player.
The hosts dared to attack on occasion, with Kyle Naughton close to becoming the second ex-Spurs player to score against his former employers as his deflected shot fizzed wide.
But it was Swansea’s defensive effort which provided the foundation for their admirable display, and looked set to earn them a first league win over Spurs since 1982.
However, their resistance was eventually broken and, with relegation rivals Hull beating Middlesbrough, the Swans’ descent back into the bottom three leaves their hopes of survival in doubt.
