The best chance in a tight first half fell to Burnley's Johann Berg Gudmundsson, who fired Aaron Lennon's cross over from close range.
Gudmundsson went close again after the break, his powerful shot forcing a fine save from Lukasz Fabianski, before Tammy Abraham had a deft flick saved by Nick Pope.
The match appeared to be heading towards a stalemate but in the 81st minute, Ki broke the deadlock when he converted Kyle Naughton's pass with a low strike from 18 yards.
Swansea's third win in four matches moves them up to 15th while Burnley, who are winless in 10, remain seventh.
Swansea: 24 Feb v Brighton (A)
Burnley: 24 Feb v Southampton (H)