Spurs brought Dele Alli back after his suspension for Wednesday’s Champions League win over Borussia Dortmund while the visitors included Mike van der Hoorn as they switched to a back three for the game.
The Lilywhites wasted little time in setting the tempo and forced an early save from Lukasz Fabianski when Harry Kane went close with a long-range effort.
It remained all Tottenham but clear chances were few and far between, with Fabianski remaining alert but untested and Van der Hoorn could even have sent his side in ahead at the break had he kept a header down from a corner.
Mauricio Pochettino’s men continued to push in the second half, having a penalty shout against Martin Olsson turned down, while Heung-min Son also forced a save from the in-form Swans goalkeeper.
It seemed only a matter of time when Harry Kane hit the crossbar from the edge of the six yard area and the England man was again thwarted after the hour mark when the brilliant Fabianski produced another reflex save to tip clear a glancing header.
Alli should have opened the scoring when he miscued a free header from close range and, as the game wore on, the tension grew around Wembley as Spurs searched for their maiden Premier League win at the ground.
Serge Aurier should have been awarded a penalty in the final minutes when tripped by Jordan Ayew but referee Mike Dean wrongly adjudged the Ivorian had handled during the build-up.
Grimsby are open to talking to Sol Campbell about their vacant managerial position - and the former England defender wants to make it happen.
Grimsby are open to talking to Sol Campbell about their vacant managerial position.