The South Korean struck with nine minutes left, his low effort creeping through a number of legs and past Clarets keeper Nick Pope to nestle into the bottom corner of the net.
Ki's goal sealed a third league home victory in a row for the Swans and extended Burnley's winless run to 10 matches.
The Clarets were quickest out of the starting blocks in the first half and enjoyed the first chance of the game on 11 minutes.
The visitors threatened again seven minutes later when Charlie Taylor's cross into the box was punched clear by Lukasz Fabianski, but Barnes was on hand to reclaim the ball outside the box before firing an effort onto the net.
Just after the half-hour mark, Aaron Lennon lofted a cross into the box which Vokes flicked on into the path of Johann Berg Gudmundsson, who in turn shinned his volley over the bar from a tight angle.
Neither side could find the breakthrough in the first half and little changed after the restart, much of the second period passing by with neither side especially dangerous in the opposition's final third.
However, Swansea dominated possession and grew into the game, with the deadlock finally broken on 81 minutes when Kyle Naughton cut the ball back to Ki, who placed a low shot beyond Pope and into the bottom-left corner.
SWANSEA CITY (5-4-1): Fabianski; Naughton, van der Hoorn, Fernandez, Mawson, Olsson (A. Ayew 59'); Dyer (Abraham 74'), Ki, Carroll, Clucas; J. Ayew (King 90+1')
BURNLEY (4-4-2): Pope; Lowton, Long, Mee, Taylor; Lennon (Arfield 74', Cork, Hendrick, Gudmundsson (N'Koudou 85'); Vokes (Wells 81'), Barnes