Home manager Tony Pulis made seven changes while visiting boss Harry Redknapp made nine.
Stoke's Wilson Palacios started against his former club, but was substituted soon after the hour mark following a disappointing display.
Fellow former Spurs player Peter Crouch came on in the 12th minute to replace Kenwyne Jones, who was forced off with an ankle injury, and the England striker gave a better showing than Palacios.
Jones had looked good up front alongside Cameron Jerome in a positive start by Stoke, but Tottenham had the only early sight of goal as Rafael van der Vaart had a free-kick blocked.
Pennant misses 1-1
Pavlyuchenko misses 1-1
(leads to sudden death)
Luongo misses 7-6
Stand-in Stoke captain Robert Huth headed over a Matthew Etherington corner, but chances were scarce as the rain poured down.
Spurs finished the first half well though, as Sandro's 25-yard shot was saved by Sorensen before Carroll volleyed over after Roman Pavlyuchenko's cross from the right.
The visitors also started the second half strongly as Van der Vaart shot wide then Livermore did the same. Pavlyuchenko's 20-yard hit was pushed over by Sorensen and another Van der Vaart free-kick came close as Spurs' strong spell continued.
Etherington then put in a cross that troubled Spurs close to the hour mark, as Gomes flapped at the ball under pressure from Crouch, but Sandro - sporting blue highlights in his hair - cleared acrobatically.
A Jermaine Pennant free-kick was deflected over the top as Stoke continued to fight back.
Jermain Defoe had a near miss with a long-range strike shortly after he came on for Van der Vaart. At the other end, Crouch continued to cause problems as the game looked like it could go either way in the final stages.
There was an unusual incident in the 87th-minute when Livermore, injured off the pitch, crawled back on to try to halt play before Jon Walters hauled him off.
Huth shot from the edge of the box early in extra time and Gomes spilled the ball then re-gathered before Crouch could pounce. The hosts missed an even better chance when Walters shot wide when one-on-one with the Spurs goalkeeper.
The teams naturally looked tired in the second period of extra time, creating few openings, in a low-key end to a poor-quality match.
In the penalty shoot-out, Walters and Defoe both scored first up for their sides. Pennant hit the bar but Pavlyuchenko also blazed over.
All the following penalties were then successful up to 7-6, when Stoke's Danish goalkeeper saved from Luongo.
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