by Graham CLUTTON
With just two minutes left, Mauricio Pochettino’s team were in danger of falling 10 points behind Premier League leaders Chelsea, who were on the verge of wrapping up a 2-1 victory against Manchester City.
Spurs were trailing to an 11th minute goal from their former winger Wayne Routledge.
However, the north Londoners refused to surrender and Dele Alli equalised before South Korean forward Son struck in stoppage-time.
Christian Eriksen made it three to ensure Spurs still have a chance of overhauling Chelsea with eight games to play.
Swansea slump back into the relegation zone and are now two points from safety.
Spurs might have taken the lead inside the first 40 seconds when Son was teed up on the edge of the Swansea box, but Federico Fernandez blocked his shot and Alfie Mawson completed the clearance.
Having been somewhat fortunate to survive an early onslaught from the visitors, Paul Clement’s side took an 11th minute lead.
Jordan Ayew forced his way into the box and duly set up Routledge to tap in.
Tottenham responded by dominating possession, but they couldn’t find a way through in the first half.
When Ben Davies crossed for Son to volley over, it was beginning to look like one of those nights when nothing went right for Spurs.
Pochettino’s side felt they should have had a penalty two minutes into the second half when Son’s goal-bound shot struck Fernandez, but replays showed the ball had hit the Argentine on the shoulder.
Swansea continued to drag all but Ayew behind the ball and while Tottenham kept their composure there were precious few opportunities to score.
It was almost a different story altogether at the other end when Kyle Naughton cut back inside Davies before seeing his deflected shot fly wide of Michel Vorm’s far post.
From the ensuing corner Leroy Fer was adjudged to have fouled the former Swansea goalkeeper in trying to head home from three yards.
Pochettino’s response was to introduce Vincent Janssen on the hour, with Alli moving further forward as well
With the game edging towards full time, Tottenham kept pressing and Toby Alderweireld saw a shot fizz over the top of Lukasz Fabianski’s goal before the Pole kept his side in front with a wonderful save from Alli’s 85th minute header.
The breakthrough finally came with three minutes remaining.
Eriksen’s shot was deflected to the far post, where Alli side-footed the ball past Fabianski, sparking wild celebrations on the Tottenham bench.
There was even greater jubilation to come for Spurs in stoppage-time.
With a win vital to their chances of catching Chelsea, the visitors poured forward and Son bagged the second after an exquisite flick from Janssen.
Swansea were shattered and Eriksen wrapped up the epic finale when the Dane beat Fabianski from 20 yards after Alli’s wonderful assist.