Australian teenager Harry Kewell had rescued Leeds with his first senior domestic goal as the tie failed to reach a conclusion inside normal time, Stoke having gone ahead with a disputed penalty midway through the second half.
But the home side faded in the extra period, enabling Leeds to ease through to the fourth round and a home tie against Reading.
Kewell played in the absence of suspended Jimmy Floyd Hasselbaink as Leeds, who would be happy to take any trophy these days, treated the competition with unqualified respect by fielding an otherwise full-strength team.
They were wise to do so against a team who had won six matches in seven. Stoke were weakened by the absence of leading scorer Peter Thorne, otherwise they might well have had the edge.
There was a good deal of caution from both sides, although Leeds, whose midfield quartet rarely allowed much space ahead of their back four, stepped up the pace in the last third of the opening half. Rod Wallace and Bruno Ribeiro each missed the target narrowly from long range.
Even then, they almost shot themselves in the foot on the stroke of half- time when defender Gunnar Halle's attempt to head back to goalkeeper Nigel Martyn was a goalpost's width from an own goal.
There was little to divide the teams and Gerry McMahon missed with a glancing header from Kevin Keen's cross just before Stoke went ahead 21 minutes into the second half.
The deadlock was broken by a penalty contentious enough to earn David Wetherall a yellow card after full-back Andy Griffin tumbled under his challenge, the Leeds defender pushing the Stoke man in the chest after claiming Griffin dived. Graham Kavanagh beat Nigel Martyn confidently.
Four minutes later, however, Leeds restored parity with an astonishing strike by Kewell, whose 25-yard left-foot drive gave Carl Muggleton no chance.
For Leeds to have won in normal time would have been rough on the home side but, with both front men giving way to inexperienced replacements, they tired in extra time and Leeds needed only two minutes to go in front, Wallace getting under Gary Kelly's right-wing cross to send a header looping out of Muggleton's reach.
At the end of the first period, Wallace arrived at the near post as Kelly delivered a cross from the left following a short corner, the striker nodding home his seventh goal of the season.
Stoke City (4-4-2): Muggleton; Pickering, Sigurdsson, Tweed, Griffin; Keen, Kavanagh (Whittle, 109), Ray Wallace, Forsyth; Andrade (Nyamah, 74), McMahon (Crowe, 85).
Leeds United (4-4-2): Martyn; Halle, Wetherall, Radebe, Robertson; Kelly, Hopkin (Bowyer, 105), Haland, Ribeiro (Lilley, 74); Rod Wallace, Kewell. Substitute not used: Molenaar.
Referee: P Jones (Loughborough).Reuse content