Swansea took the lead through Andre Ayew's 17th-minute header, but Christian Eriksen got us back on level terms with a free-kick 10 minutes later.
An unfortunate Harry Kane own goal gifted the advantage back to the hosts, but we responded once more through another Eriksen free-kick.
We pushed and pushed in search of a winner but were unable to beat Lukasz Fabianski for a third time, while a fantastic save from Hugo Lloris deep into stoppage time - the keeper turning Federico Fernandez’s header onto the crossbar - ensured we took a point from our trip to south Wales.
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.