Christian Eriksen scored two sublime free-kicks as we came from behind twice to gain a point at Swansea – and were denied all three by Lukasz Fabianski.
The goalkeeper made save after save to keep us at bay – post-match stats showed he made nine in total – as we racked up 15 shots to Swansea’s seven.
Eriksen’s fine second goal made it 2-2 in the 64th minute and we threw everything at the home side in the final half-hour but couldn’t find a way past the Polish stopper.
Incredibly though we could have found ourselves on the end of a 3-2 defeat in the final minute of added time as Hugo Lloris made his one and only save of note – and arguably the best of the game – to tip Federico Fernandez’s header onto the crossbar.
There were two changes from the team that drew in Monaco on Thursday – both at the back – with Kyle Walker and Ben Davies coming in for Kieran Trippier and Danny Rose respectively.
Meanwhile on the bench it was good to see Mousa Dembele back in the squad following his injury - he would be introduced in the second period.
It felt distinctly autumnal on the south Wales coast as the Premier League season rolled into October with only the brave opting for short sleeves in the stands of the Liberty Stadium.
We started the brighter in the opening exchanges and had to cope with a well-watered pitch which made it heavy going at times.
Toby Alderweireld looped a header over from an Erik Lamela corner but that was the only chance of note inside the opening 10 minutes before Nacer Chadli tested the hands of Fabianski from distance.
Our pressing style was causing problems for the home side as we won several early turnovers of possession.
We had looked in complete control yet it was the home side who broke the deadlock against the run of play through Andre Ayew. Jefferson Montero picked out his run following a good counter and he rose to head past Lloris from eight yards.
The Swans had looked flat but the goal seemed to give them life and turned the atmosphere up within the Liberty Stadium.
Chadli had our best chance midway through the first period when he cut in from the left and forced a smart stop from the goalkeeper before the rebound was scrambled clear, but the equaliser was not far off.
Eriksen went close to scoring from a free-kick last weekend when he rattled the woodwork against Manchester City but he went one better this time, bending a set piece up, over the wall and past the goalkeeper from just outside the box on the left after Dele Alli was upended by Fernandez.
It was what we had deserved and our number 23 almost grabbed another goal two minutes later but dragged his shot just wide following a good break.
The game had come to life and the home side soon grabbed the lead once more, albeit with a large slice of luck. Jonjo Shelvey took a corner from the left which failed to clear the first man but as Harry Kane went to clear it, the ball sliced off his boot and beyond Lloris.
Kane almost made amends soon after when he was fed by Lamela and forced a good low save from Fabianski on his left foot.
You daren’t take your eyes off the action as Eriksen was a whisker away from scoring through another free-kick, although Fabianski this time palmed his effort away.
We saw plenty of the ball in good areas and with five minutes of the half remaining Fabianski kept his side ahead as he again palmed behind a decent effort from Eriksen.
We ended the first period camped inside the home side’s half but unable to force a leveller.
The second half started in the same fashion and Kane almost got us level within a minute of the restart but couldn’t poke his effort past Fabianski after a good ball in from Davies.
Bafetimbi Gomis might have done better with a header shortly afterwards as both sides pushed for the next goal. It was entertaining stuff.
The frantic tempo simmered down into a more balanced rhythm as the game progressed, however, with both sides retaining the ball better and Gomis becoming more of a threat for the home side.
Swansea were in the ascendancy but were stopped dead in their tracks once more by the brilliance of Eriksen, who levelled the scores with 25 minutes to go. It was a free-kick in the same area from which he scored the first, but this time he whipped it across the keeper and into the top corner.
Andros Townsend replaced Lamela and gave us an outlet and fresh legs on the right soon afterwards, while Alli almost put us ahead for the first time in the game with 15 minutes remaining when he stung the palms of Fabianski from distance.
Townsend was next to go close as he cut in from the right and unleashed a shot that again Fabianski managed to save, this time with his feet.
Townsend knocked on the door again and Eriksen was inches away from a memorable hat-trick in the 90th minute, but it was then left to Lloris to preserve our point in added time when he pushed Fernandez’s header onto the bar. A share of the spoils was the very least we deserved from this display.
Swansea (4-2-3-1): Fabianski, Rangel, Williams (c), Fernandez, Taylor; Shelvey (Cork 86), Sung-Yueng; Ayew, Sigurdsson (Barrow 74), Montero; Gomis. Substitutes (not used): Nordfeldt, Tabanou, Routledge, Eder, Bartley.
Spurs (4-2-3-1): Lloris (c), Walker, Alderweireld, Vertonghen, Davies; Alli, Dier; Chadli (Njie 63), Eriksen, Lamela (Townsend 57); Kane (Dembele 80). Substitutes (not used): Vorm, Trippier, Wimmer, Carroll.
Goals: Swansea - Ayew 17, Kane (own goal) 31. Spurs – Eriksen 27, 64.
Yellow cards: Swansea - Fernandez 26. Spurs – Davies 60, Dier 79, Vertonghen 83, Walker 89.
Referee: Mike Dean.