Spurs have now dropped seven Premier League points at the ground this season, more than they surrendered at White Hart Lane throughout the whole of the 2016-17 campaign.
Mauricio Pochettino's side claimed 66% of the possession during the first period, and while they created chances, they were not clinical enough in front of goal.
The Whites came closest to breaking the deadlock in the first half on 11 minutes when Harry Kane tested Lukasz Fabianski with a powerful free kick, the Polish goalkeeper turning the ball behind for a corner.
The story was the same after the interval, with Spurs keeping up the pressure and Swansea absorbing it, though the hosts were inches from going in front just before the hour mark when Moussa Sissoko picked out Kane in the box and the England international rattled the crossbar.
Paul Clement's men defended tightly but created several chances at the other end, coming closest to hitting Spurs on the break in the 78th minute when Sam Clucas's volley flew narrowly over the crossbar.
There was controversy two minutes from time when referee Mike Dean failed to spot a foul on substitute Serge Aurier in the box, Jordan Ayew going unpunished and Toby Alderweireld receiving a yellow card for his protests.
In midweek, Spurs hit three at Wembley during a convincing win over Borussia Dortmund in the Champions League, but they were unable to repeat the feat against a tenacious Swansea side.