Swansea completed the comeback shortly after the hour mark when substitute Adam Idah scored with his first touch, flicking a cross into the net in the 66th minute.
Stoke pushed for an equaliser during the closing stages but could not find a way through, leaving Swansea to celebrate an impressive opening-day victory.
For Stoke, the defeat will be a frustrating start to the Championship campaign, particularly after having taken an early advantage, while Swansea will take plenty of confidence from coming from behind to secure all three points away from home.