Jonathan Walters and Stephen Ireland gave the visitors the half-time advantage.
Wilfried Bony got one back early in the second half before Nathan Dyer levelled and the Ivorian struck his 10th goal of the season to put Swansea ahead with four minutes to go.
But Swansea, who had let leads slip in the dying moments of both Europa League meetings with Kuban Krasnodar, were denied again when referee Robert Madley decided Wayne Routledge had handled a Ryan Shawcross header and Adam made no mistake from the spot.
Such a result had seemed unlikely after a lacklustre first-half display from Swansea, but the shell-shocked Welsh side will feel aggrieved to not have taken all three points after a rousing fightback.
Stoke, meanwhile, see their winless run extended to eight games and manager Mark Hughes will be furious they contrived to throw away a fully-deserved interval lead.
The visitors had hit the front in just the eighth minute.
Ashley Williams was off the field after treatment, desperately trying to come back on, when Peter Crouch flicked on a long ball and Ben Davies and Chico Flores got in each other's way to allow Walters to apply the finish.
Williams was furious and let his frustrations show with a poor challenge on Steven Nzonzi for which he was lucky not to be booked.
Swansea could have levelled in the 15th minute but Roland Lamah headed straight at Asmir Begovic after good work by Alejandro Pozuelo and Angel Rangel.
But instead they fell further behind with 25 minutes gone. Leon Britton was robbed of possession and two lovely passes from Walters and Crouch ripped Swansea apart for Ireland to slot beyond Tremmel.
Flores, culpable in both Stoke goals then disgraced himself with an embarrassing attempt to get Nzonzi sent off.
The midfielder was involved in an exchange of words with Bony and pushed out at Flores' chest, but the Spaniard collapsed in a heap clutching his face and was rightly booked.
To compound Swansea frustrations they then spurned a gilt-edged chance to pull a goal back.
Erik Pieters' back pass found a lurking Bony, but Begovic knocked the ball away from the striker as he attempted to round the keeper, before saving Routledge's follow-up.
The Ivorian made no such mistake in the 56th minute. Stoke failed to clear a Jonathan de Guzman corner and the Dutchman did not waste the second chance as he picked out Bony, who powered in his header from close range.
Swansea suddenly discovered a flow to their game and an exquisite exchange of passes between Britton, Pozuelo and Routledge ended with Bony's attempted backheel being deflected wide.
Stoke were having to deal with incessant pressure but could not hold out.
Routledge surged past Geoff Cameron to the byline and when Walters' headed clearance dropped to Dyer, the substitute volleyed home with the aid of a slight deflection.
With the game on a knife-edge, frayed tempers began to boil over and Rangel was lucky not to see red after slapping Crouch in the midst of a touchline fracas.
Bony then struck what looked like being the winner, slotting home left-footed from Jonjo Shelvey's pull-back, but Stoke emerged with a point in the late drama.
Source: PA
Source: PA