The first chance fell to Leicester when Lloyd Dyer's 12-yard shot from a Richie Wellens cross was blocked by Gary Monk.
Steve Howard then drove over as the home side enjoyed the better of the opening exchanges.
But Swansea appeared to have the Foxes on the run when on 16 minutes they gifted a free header to Ashley Williams from Mark Gower's corner and the visitors took the lead.
It was the highlight of a drab first half which ended with Matt Fryatt threatening to latch on to a Wellens threw ball, but Gary Monk snuffed out the danger.
The home side were thrown a lifeline on 52 minutes when Bruno Berner was tripped inside the box by Ashley Williams.
But they failed to haul themselves level when Dorus de Vries flung himself to the left to push away Steve Howard's spot-kick.
De Vries rescued Swansea again 67 minutes when he reacted sharply to keep out Michael Morrison's close-range effort.
But the game was turned on its head with two Leicester goals in as many minutes.
First on 69 minutes Berner crossed low to Martyn Waghorn, who drove home a deflected shot beyond the wrong-footed de Vries.
Then on 71 minutes Howard got a head to a Matty Oakley corner for Dany N'Guessan to drive in the ball from close range.
And the home side superiority was almost rewarded with a third goal when N'Guessan crashed a header against the underside of the bar, before fellow half-time substitute Waghorn saw his drive drift just wide of the far post.