Former Spur Wayne Routledge's early close-range finish gave the home side the lead in front of a raucous crowd at the Liberty Stadium and from then on we were frustrated for long periods as we sought to break down a packed defence.
Łukasz Fabiański saved well from substitute Vincent Janssen and Dele Alli with just under 10 minutes to go - perhaps signalling that it wasn't going to be our night in South Wales.
But what character this team has!
With changes made, Dele scrambled home an 88th-minute equaliser and immediately you got the feeling we could grab a winner as we carried on pushing with seven minutes going up on the fourth official's board.
And sure enough, Janssen's touch set up Heung-Min Son to slip beyond Fabianski to put us sensationally ahead in the opening 60 seconds of added time.
There was still more to come, though. Janssen was involved again as Dele sent Christian Eriksen through to complete a 3-1 victory on the counter-attack with 94 minutes played.
It was mightily close, but we remain undefeated against the Swans in the Premier League era, with the three points allowing us to keep up the pace with Premier League leaders Chelsea, who won 2-1 at home to Manchester City.
Mauricio Pochettino made five changes from the side that was victorious at Burnley on Saturday. Hugo Lloris missed out after coming down unwell, meaning Michel Vorm stepped in against his former club with Jan Vertonghen taking over as captain. Kyle Walker returned at right-back with Mousa Dembele, Moussa Sissoko and Son all coming into the starting line-up after appearing from the bench at Turf Moor. The Swans had four of our ex-players in their starting XI - Routledge, Gylfi Sigurdsson, Tom Carroll and Kyle Naughton.
GOAL: SWANSEA 1-0 SPURS - WAYNE ROUTLEDGE – 11mins
We had started brightly, creating two good chances in the first five minutes, but the home side wrestled back the advantage with a series of corners and grabbed the lead in direct fashion. A thumping clearance upfield from goalkeeper Fabianski bounced behind our defensive line for Jordan Ayew to run onto. He took it all the way to the byline before cutting it back for Routledge to bundle home from close range.
CHANCE: SON FIRES OVER - 42mins
Despite going behind, we dominated possession but found ourselves up against a resilient, deep-lying Swansea rearguard that proved hard to penetrate. After a number of half-chances, we created a decent opening as the half ticked on as Eriksen unlocked the defence with a perfectly-weighted through ball for ex-Swansea man Ben Davies who cut back for Son, but the forward cushioned a volley well over the bar.
CHANCE: NAUGHTON ALMOST MAKES IT 2-0 – 56mins
The second half started in more even fashion and we had shouts for a penalty fall on deaf ears just two minutes into the half when Son appealed for a handball against Federico Fernandez. Swansea were next to go close though when Naughton skipped inside from the right and fired a shot across goal that took a slight deflection on its way behind for a corner. Following that it was all Spurs, however chances remained at a premium as the home side defended deep inside their own box.
Above: Dele Alli up against Wayne Routledge.
SAVES: FABIANSKI DENIES JANSSEN AND DELE - 80-85mins
Our first clear chance of the second half arrived with nine minutes remaining as Janssen, on for Sissoko, fashioned a one-two with Dele, going in for a brave 60-40 and toe-poking the ball at goal, blocked by Fabianski. The keeper took a whack but carried on after treatment and the sight of the stretcher coming on. Toby Alderweireld fired over before Fabianski made a flying save to his left to keep out Dele's 85th-minute header from Eriksen's cross.
GOAL: SWANSEA 1-1 SPURS - DELE ALLI - 88mins
We finally found a breakthrough with two minutes of normal time left. Eriksen's half-volley from the right-angle wasn't dealt with by the Swans and it deflected through for Dele to tap in at the far post.
GOAL: SWANSEA 1-2 SPURS - HEUNG-MIN SON - 90+1mins
Straight after the equaliser, Janssen tested Fabianski with a snap-shot before we dramatically took the lead in the first minute of stoppage time. Eriksen played the ball in to Janssen, whose lovely touch took it into the path of Son, the South Korean international duly squeezing the ball home under Fabianski.
GOAL: SWANSEA 1-3 SPURS - CHRISTIAN ERIKSEN - 90+4mins
We made sure of the victory three minutes later as Janssen found Dele, who played a fantastic pass into the path of Eriksen, who then shimmied wide of Martin Olsson before picking his spot from 20 yards and producing a clinical finish. Alfie Mawson went close for Swansea moments later but the points were ours.
VINCENT JANSSEN SAID
"It was late and dramatic but it was very important. If you see the other results then it’s nice that we got the three points here and in the end I think we deserved it because we were attacking all game. They scored one goal and just defended and in the end we broke their defence open and scored three goals. I was happy that I came on and could help the team. Of course I’m happy that we won and it was great to set up the goal for Sonny. I think he finished it well and the last goal was good as well."
MAURICIO POCHETTINO SAID
"Everything can happen if you never give up and you believe, have faith and keep pushing. Today was a good example of how the team showed character and personality and how we fight for the badge and that is most important. We can be proud of the supporters and the supporters should feel the same about the team."
Swansea (4-3-3): Fabianski; Naughton (Sung-Yueng 72), Fernandez, Mawson, Olsson; Carroll, Fer, Cork (c); Routledge (Narsingh 90+4), Ayew (McBurnie 75), Sigurdsson. Substitutes (not used): Nordfeldt, Amat, Montero, Kingsley.
Spurs (4-3-2-1): Vorm; Walker, Alderweireld, Vertonghen (c), Davies (Nkoudou 79); Dier, Dembele; Sissoko (Janssen 61), Dele, Eriksen; Son (Trippier 90+4). Substitutes (not used): Pau Lopez, Carter-Vickers, Wimmer, Onomah.
Goals: Swansea - Routledge 11; Spurs - Dele 88, Son 90+1, Eriksen 90+4.
Yellow cards: Swansea - Ayew 18; Spurs - Dier 79.
Referee: Jonathan Moss.
We are back at White Hart Lane for the early kick-off (12.30pm) on Saturday with the visit of Watford. The Hornets arrive in good form off the back of two wins, although we will be hoping for a repeat of our last encounter when claimed a 4-1 win at Vicarage Road at the turn of the year.