The Potters triumphed in the shoot-out on Tuesday night thanks to Thomas Sorensen's save from Spurs debutant Massimo Luongo.
The game had finished goalless after extra-time, with Sorensen palming a powerful strike by Roman Pavlyuchenko over the bar in the 53rd minute and Stoke substitute Jon Walters failing to convert a one-on-one chance in the additional half-hour.
Although both teams were much-changed from their previous outings, it seemed the fact they were each playing for the third time in six days due to their Europa League exploits took its toll, particularly in the first half, with neither showing much dynamism going forward.
Stoke suffered an early setback when Kenwyne Jones pulled up with an injury in the 10th minute and had to limp off.
Former Tottenham man Peter Crouch came on to take his place and moments later, Robert Huth - Potters captain for the night in the absence of Ryan Shawcross - nodded the first real chance of the contest off target from Matthew Etherington's corner.
Etherington then delivered a cross into the danger zone which came to Crouch, but the England striker was unable to get proper control on his header and Spurs cleared.
The two players combined again just after the half-hour mark with Crouch knocking down Etherington's ball into the box, but Younes Kaboul was on hand to avert the danger.
Sorensen got down to save a shot from outside the area by Sandro and Tottenham youngster Tom Carroll hit a volley over the bar.
Etherington then tried his luck from distance but could only send his effort bending wide.
Spurs started the second half brightly, with Jake Livermore curling the ball just the wrong side of the post and Pavlyuchenko smashing a fierce drive goalwards which Sorensen pushed over the bar.
Having attempted to squeeze through and been brought down by Huth on the edge of the box, Rafael van der Vaart struck the resulting free-kick wide.
At the other end, Heurelho Gomes did well to punch the ball away as Crouch and Cameron Jerome tried to force in Etherington's cross.
Crouch then made insufficient contact to divert another Etherington delivery past Gomes.
Ryan Shotton was next to whip the ball into the Tottenham area, Kaboul managing to hook it away.
Jermaine Pennant got the better of Benoit Assou-Ekotto on the right flank and cut into the box with five minutes of normal time remaining, but the Potters midfielder took a touch too many and could only win a corner which amounted to nothing.
Stoke continued to press, but a collision between Kaboul and substitute Salif Diao slowed the tempo again and referee Lee Probert blew his whistle to confirm the need extra-time.
The hosts went close soon after the action resumed, Gomes spilling Huth's drilled effort but then reacting to save Crouch's follow-up.
A quick free-kick just before half-time in extra-time slipped Walters - who had come on for Jerome - in on goal, but he dragged his shot wastefully wide.
Carroll hit an attempt on the turn that flew off target early in the second additional period and Etherington fired over in the final moments.
Sorensen proved Stoke's hero in the penalty shoot-out that followed, stopping Luongo's spot-kick to seal the Potters' progress to the last 16.