Alex Neil's Stoke City reign began with a draw thanks to a stoppage-time goal from Tyrese Campbell.
But substitute Campbell swept in to make it a decent night for Neil, who was in the dugout for the first time since being named Stoke boss.
His new side climb to 16th in the early-season table, while Swansea edge out of the drop zone into 21st.
The Welsh club thought they had done enough to secure what would have been a pressure-easing victory, but the timing of Campbell's leveller was hard to take.
Despite a much-improved performance, a return of only one victory from Swansea's first seven Championship games in 2022-23 is a concern for head coach Russell Martin.