When we finally looked to have cracked it, goalkeeper Begovic produced a world-class save in the 90th minute to palm away Sigurdsson's header low down to his right.
Sandro delivered the cross from the right and It looked for all the world that Sigurdsson's header was going to nestle in the back of the net from 12 yards, but it wasn't to be.
Andre Villas-Boas made three changes from last week's win against Swansea.
Michael Dawson and Gareth Bale returned from injury and were both back in the team with Dawson partnering Steven Caulker in central defence, the third change.
The game played in pouring rain almost started disastrously as Kyle Walker looked to switch play from right to left but Kenwyne Jones intercepted on the edge of our box, in a central position. Jan Vertonghen threw himself in to block as the striker fired well wide - less than 30 seconds on the clock.
Stoke had another chance on 15 minutes when Hugo Lloris managed to touch Jones' effort away and Sandro was there to clear the loose ball from Matthew Etherington.
Vertonghen's influence was certainly further up the field for the rest of the match and he delivered a string a fine crosses to no avail in the first half.
The full-back somehow got away from Ryan Shotton, Jon Walters and Geoff Cameron in the 20th minute and then checked back onto his right foot to deliver a fine cross that Jermain Defoe was inches away from nodding home.
Mousa Dembele sent Bale away in the 34th minute and the winger crossed deep for Emmanuel Adebayor, who stretched to loop a header across Begovic but onto the roof of the net. Dembele and Defoe went close from distance just before half-time.
The second half followed a similar pattern as we tried to break down the Potters' defence.
Defoe got to the byline but crossed behind Adebayor and Aaron Lennon and then Bale rose well to meet Lennon's cross at the far post but couldn't keep his header down.
At the other end, Ryan Shawcross might have done better with a free header from Etherington's free-kick and the same could be said of Shotton's somewhat tame shot from Cameron's cross.
We continued to push on for that one clear-cut chance. Defoe's clever chipped pass released Sandro on the right, but he chose to cross and the chance was cleared. Begovic then did well to hold onto a tricky dipper from Bale.
The last two chances arrived either side of the 90th minute.
Adebayor weighted the perfect pass between defenders to Bale, who looked set for a first-time strike with his left foot. However, substitute Cameron Jerome tracked him all the way and just did enough to throw Bale off balance as he went to shoot.
Sigurdsson's chance arrived 45 seconds into the additional four minutes, directing Sandro's flat cross on target, but instead of the net bulging, Begovic got down to preserve another clean sheet.
Spurs (4-4-2) - Lloris; Walker, Caulker, Dawson, Vertonghen; Lennon (Sigurdsson, 78), Dembele (Parker, 85), Sandro, Bale; Adebayor, Defoe. Unused subs: Friedel, Gallas, Naughton, Livermore, Townsend.
Stoke City (4-2-3-1): Begovic; Cameron, Huth, Shawcross, Wilkinson; N'Zonzi, Whelan; Shotton (Whitehead, 64), Walters, Etherington (Crouch, 69); Jones (Jerome, 66). Unused subs: Sorensen, Palacios, Upson, Kightly.
Yellow cards: Spurs - Sandro, Vertonghen. Stoke - Wilkinson, Whitehead, Cameron, N'Zonzi, Walters.
Referee: Mr L Mason.
The defensive resilience that has taken Stoke up to eighth in the Premier League - conceding just 13 goals along the way - was there for all to see as we knocked and knocked, but couldn't find a way through a wall of blue and crimson stripes.