Their winner from Wayne Thomas came three minutes into time added on at the end of the game, after Wimbledon looked to have snatched a point.
Stoke, who were composed and hardworking throughout, went in front from the penalty spot after Darel Russell was brought down in the area by a clumsy double challenge by Alex Tapp and Nigel Reo-Coker.
Carl Asaba fired the spot-kick into the top corner of the net sending former Stoke keeper Steve Banks the wrong way.
Just minutes later Reo-Coker had the chance to make amends and level the scores of blasted high over the bar from close range.
Then in the 34th minute Gifton Noel-Williams who had hit the post in the fifth minute raced on to a free ball only to see Banks save bravely at his feet.
Stoke were stunned when the visitors pulled level ten minutes into the second half. Reo-Coker split the Stoke defence with a superb through ball for Patrick Agyemang who fired past the oncoming keeper Neil Cutler.
Minutes later Reo-Coker almost hit a second for Wimbledon but Cutler produced a super save to deny him.
Stoke piled on the pressure and in the dying minutes Banks produced a magnificent save to deny substitute Kris Commons.
But Stoke were not finished and defender Thomas grabbed the winner in time added on after Noel-Williams had knocked the ball on in the penalty area.
Stoke boss Tony Pulis said that: "It was a good open game. Wimbledon have a lot of young players and are a very vibrant side. They created three great opportunities but we were a threat for 90 minutes.
"No one is going to get carried away and as yet we are still 44 points away from safety. We have a very young group of players they have done well so far and I am delighted for them."
Wimbledon boss Stuart Murdoch said: "The late save by Steve Banks was magnificent and I thought that he had saved us a point.
"The players are absolutely gutted after being beaten by a such a late goal, I thought they deserved a point.
"The team showed a lot of character and after a bad result in midweek they did well to come here and put on such a performance."