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.
Here is how the action unfolded with Sports Mole's live match commentary.
2.00pmHello and welcome to Sports Mole's live coverage of the Premier League meeting between Swansea City and Burnley.
2.02pmAt first glance this appears to be a clash between a high-flyer and a relegation fighter, but the two sides' respective forms are very different indeed.
2.04pmThe Swans continue to impress under Carlos Carvalhal, claiming 11 points from six league matches under him - including wins over Liverpool and Arsenal - and recording an emphatic 8-1 victory over Notts County in their midweek FA Cup fourth-round replay.
2.06pmBurnley, meanwhile, are without a Premier League victory in nine games, the longest current winless run in the division, and if things don't go their way today, could remain winless in their opening six league matches of a calendar year for the first time since 2007.
2.08pmTeam news have filtered through so let's have a look at how these two sides will line up today at the Liberty.
2.09pmSWANSEA STARTING XI: Fabianski, Ki, van der Hoorn, Mawson, Dyer, Carroll, Olsson, Clucas, J. Ayew, Naughton, Fernandez
2.14pmLet's start with the hosts, and Swans boss Carlos Carvalhal has made six changes from the rotated side that beat Notts County 8-1 in the FA Cup replay in midweek.
2.15pmLukas Fabianski returns in goal, while fellow regulars Martin Olsson, Federico Fernandez, Alfie Mawson, Ki Sung-yueng and Jordan Ayew are also drafted back into the starting XI.
2.16pmKristoffer Nordfeldt, Kyle Bartley, Luciano Narsingh, Wayne Routledge and Tammy Abraham drop to the bench, while Connor Roberts is out of the squad.
2.18pmAndre Ayew, the club's record re-signing, is on the bench alongside fellow new arrival Andy King, while Nathan Dyer hits a milestone as he prepares for his 300th Swansea appearance.
2.20pmJust the one change for Burnley after last week's 1-1 draw with Manchester City as Matthew Lowton comes back in for the injured Phil Bardsley.
2.22pmCarlos Carvalhal certainly appeared an odd choice to take the reins at Swansea, given he had not long left a club in the division below, but his appointment seems very much justified already.
2.24pmThe Swans appeared very much down and out when the Portuguese came on board but 11 points from six games have galvanised the team, now sitting 17th in the table having navigated several difficult games which will give them a huge psychological boost going forward.
2.26pmThey welcome the Clarets at the Liberty hoping to extend their unbeaten run in all competitions to nine games, and the midweek thumping of Notts County, a team that gave them a tough test in the first FA Cup game at Meadow Lane, will also have done their confidence a world of good.
2.28pmBurnley's form has stuttered of late, but they still sit seventh in the table having come out the other side of a tough run of fixtures, including a superb 1-1 draw with Manchester City at Turf Moor.
2.30pmBurnley last won on December 12 last year, beating Stoke City 1-0, but they've been putting some solid performances nonetheless, so will today be the day that they can finally get back to winning ways?
2.32pmLet's have a look now at what both managers have said in the build-up to the game, starting with the Swans boss.
2.34pmCarvalhal told reporters at his pre-match presser: "The situation is still serious - but we have a better chance now of staying in the Premier League. We were the last team and five points away from safety, but in this moment we are not under the water. You put your head out and we are breathing fresh air. But the sharks are around and they can push you down again. We have to make sure these things do not happen and we stay above the water."
2.36pmDyche, meanwhile, told his press conference: "Our fans have been amazing throughout not winning as many games lately. They're seeing beyond that. As well, there's life in a side that's giving everything - and our side certainly gives everything. We haven't got that win recently but we've played some of the top sides, and that's very, very difficult."
2.38pmThe Clarets will be hoping to pull off a double against Swansea today, having won the reverse fixture on November 18. On that day, first-half goals from Jack Cork and Ashley Barnes gave the hosts all three points at Turf Moor in a 2-0 win.
2.40pmGoing into that game, Swansea had won their previous four Premier League meetings against Burnley. The overall head-to-head record between the two reads 16 Swans wins, 18 Burnley wins, and 13 draws.
2.42pmSwansea could equal their longest home winning streak against Burnley in the league - a run of four second-tier victories from 1931 to 1934.
2.44pmBurnley's last away win at Swansea, meanwhile, came in December 1995 at the Vetch Field in the third tier, winning 4-2.
2.46pmWith just under 15 minutes to go until kickoff, let's look at some more stats surrounding both teams going into the game.
2.47pmSwansea have earned seven points from losing positions in their six Premier League games under Carvalhal, beating Watford and Arsenal and drawing against Leicester City.
2.48pmThe Swans will have won three consecutive home league games for the first time since the end of last season should they prevail today, another sign of the upturn in form under the Portuguese.
2.49pmBrothers Andre and Jordan Ayew could play together in a league fixture for the first time since Marseille's 3-2 home defeat against Stade de Reims on 26 October 2013 - the former is on the bench today, the latter in the starting XI.
2.50pmA third of Burnley's 21 league goals this season have come in the final 10 minutes of matches, the highest ratio in the division - those seven goals have earned the Clarets two wins and four draws.
2.52pmBurnley have also not won any of their last 15 matches in Wales in all competitions since beating Cardiff in April 2000, the run yielding four draws and 11 defeats.
2.54pmSCORE PREDICTION: Time for me to have a go at predicting today's result in South Wales. As good as the Swans have been in recent weeks, Burnley are a tight-knit, well-drilled side with heightened confidence from drawing with Man City, having had more rest than today's hosts, who were in midweek action. So I'm going to go for a goalless draw.
2.56pmOne Premier League game has finished today, the North London derby seeing Tottenham Hotspur recording a 1-0 win over Arsenal at Wembley Stadium.
1 minKICKOFF! Andre Marriner is the ref today and he gets the action underway, Burnley kicking off!
2 minBurnley on the front foot in these early stages as Cork looks to find space but there's a trip from behind, Marriner lets play continue though.
4 minGudmundsson charges down the left wing and cuts the ball back for Hendrick on the edge of the box, but the latter's effort on goal is charged down.
6 minLittle to write home about so far, Swansea enjoying the lion's share of possession so far as they probe for an opening, the conditions at the Liberty aren't the best and the turf is wet so both sides will need to get to grips fairly quickly with their passing.
7 minClucas makes headway down the left flank and finds himself in space to attempt a cross but his delivery ends up being reclaimed by an opposition defender.
9 minFernandez completely miscues an attempted clearance, once again the wet pitch playing a part, but no Burnley player is nearby to make the most of it as the ball goes out for a throw-in.
11 minCHANCE! We have our first effort on goal and it falls to Burnley as Barnes and Vokes combine well and the latter fires on the turn from 25 yards out but the shot sails just wide.
13 minOlsson is clattered by Lennon out on the touchline in front of the managers to gift Swansea a free kick, Dyche visibly disgruntled as he feels the officials are being harsh on his side.
15 minGudmundsson crosses into the box but the effort sails comfortably into the arms of Fabianski.
17 minWe're past the quarter-hour mark and the game hasn't quite settled just yet, possession keeps switching between the two sides and neither team has enjoyed a particularly fruitful spell of action.
18 minCHANCE! Taylor crosses into the box and Fabianski punches clear with Vokes lurking, the ball is then recycled but Barnes volleys over the bar.
20 minMee hacks down Ayew with a cynical challenge and Swansea get a free kick, but it's wasted as it eventually ends up out for a throw-in.
23 minAnother quiet period with little to write about, Swansea have enjoyed around two-thirds possession but are yet to register a shot on target or create any real chances.
26 minBarnes bursts through the Swansea back line and tries to find Lennon in the box, but the hosts' defence is on hand to sweep up.
28 minBOOKING: Olsson is booked for simulation, having hurled himself to ground after Taylor stuck a leg out but didn't make contact with the Swans man.
30 minCarroll whips a cross into the box which sees Clucas connect with a header, but the effort fails to trouble Pope in the Burnley goal as the ball is easily gathered.
32 minCHANCE! Lennon lofts a cross into the box which Vokes flicks on into the path of Gudmundsson, who in turn shins his volley over the bar from a tight angle.
34 minCHANCE! Swansea now with a bright spell of pressure as Olsson flashes the ball across the face of goal with nobody in the six-yard box to convert, and moments later Ayew and Ki both reclaim possession but can't find a way through the opposition defence, which eventually clears.
36 minCHANCE! Ki threads the ball into the path of Clucas, who turns on the edge of the box and works space for a shot at goal, but he gets too much under the ball and it's blasted over!
39 minAyew is the latest Swans player to whip in a dangerous cross but once again there are no white shirts to apply the finishing touch in the box.
41 minNot long until half time now and both sides have enjoyed several forays deep into opposition territory but it has provided little in terms of goalmouth action and thus it remains goalless.
43 minVokes tries to play a one-two with Barnes as Burnley launch a counter but the wet surface causes possession to be lost and the opportunity to go amiss.
44 minGudmundsson's cross is met by the head of Vokes, but the chance is blazed over as one minute of added time is announced.
3.47pmLennon fires tamely at Fabianski shortly before the half-time whistle blows and it remains goalless between Swansea and Burnley, with neither side having mustered a shot on target.
3.49pmThere has been little to shout about in this encounter at a wet Liberty Stadium. Burnley have been the more threatening side, but Swansea have enjoyed the lion's share of possession.
3.52pmBarnes and Vokes both mustered decent chances for the visitors, while Gudmundsson had a great opportunity on the volley but he connected poorly and shinned the ball over the bar.
3.54pmPretty much the only thing of note in the opening 45 minutes was Olsson's booking for simulation, going theatrically to ground after trying to get past Charlie Taylor, who admittedly did put a leg out but did not make contact with his opponent.
3.57pmIn terms of stats, Swansea have enjoyed 60% possession, one shot off target, zero shots on tarrget and completed 259 passes, while Burnley have had 40% possession, four off-target efforts, zero shots on target and 182 passes.
4.00pmThe other Premier League matches haven't exactly provided goalfests either - Everton against Crystal Palace is also goalless, Brighton & Hove Albion lead Stoke City 1-0, West Ham United are 1-0 up against Watford and Tottenham Hotspur won 1-0 against Arsenal.
46 minKICKOFF: Andre Marriner gets the second half underway as Swansea kick off.
47 minNeither side has made any changes at the break, Swansea quicker out of the starting blocks and winning a corner but nothing comes of it.
50 minCHANCE! Barnes opts to have a go from distance but his curling effort is just wide with Fabianski sprawling.
52 minCHANCE! Vokes wins Burnley a free kick in a dangerous position which Gudmundsson swings in, the ball falls to Barnes but his header goes high and wide over the bar.
54 minVan der Hoorn lofts a good ball beyond the Burnley back four but nobody is on hand to latch onto it, quite a repetitive theme going on here at the Liberty today.
56 minCHANCE! A miscued Fernandez clearance falls to Gudmundsson, who controls the ball with his knee and then fires in a rising half-volley that Fabianski tips over for a corner. Getting closer to a goal here!
58 minBOOKING: Hendrick is cautioned for a clumsy challenge from behind on Ayew, gifting Swansea a free kick which Ki takes but Pope saves with ease.
59 minSUBSTITUTION: Swansea make the first change of the game as Martin Olsson is been replaced by Andre Ayew.
60 minThe change means that Andre and Jordan are now playing together for the first time since Marseille's 3-2 home defeat against Stade de Reims in October 2013.
62 minSwansea win a corner as Ki strikes towards goal from a tight angle and the ball comes off Taylor, but Carroll's delivery is dealt with easily.
65 minPast the hour mark now and Swansea continue to enjoy the majority of possession, as they did in the first half, but there's still little quality in the final third to speak of a goal just yet.
67 minCHANCE! Dyer fouls Taylor to give Burnley a free kick on the left-hand touchline which Gudmundsson swings into the box towards Mee, who connects but sends his header over the bar.
71 minFernandez upends Barnes to gift Burnley a free kick 25 yards from goal, which Gudmundsson takes, it comes off the face of Clucas and is subsequently punched away by Fabianski. Now we have a stoppage in play as Clucas remains down.
74 minSUBSTITUTIONS: Burnley make their first change of the afternoon, with Aaron Lennon being replaced by Scott Arfield, while Swansea make their second, Nathan Dyer off for Tammy Abraham.
76 minLowton tries to make headway down the left flank on the overlap but he's beaten to the ball by the full-back, more urgency from both sides now with the game entering its final 15 minutes plus stoppage time.
79 minCHANCE! So close to an opener from Swansea as Jordan Ayew gets the faintest of touches from an inswinging cross, but despite Pope going the other way, he sticks his hand out and keeps the ball out!
81 minGOAL! SWANSEA CITY 1-0 BURNLEY (KI SUNG-YEUNG)
82 minWith just nine minutes left in this game, Swansea break the deadlock! Ki gets the finishing touch, rifling a low strike through several legs from the edge of the box before nestling in the bottom left corner after Naughton had cut the ball back into the South Korean's path!
85 minSUBSTITUTION: Sean Dyche opts to bring on Georges-Kevin N'Koudou in place of Johann Berg Gudmundsson.
87 minCHANCE! Nearly a second for Swansea as Jordan Ayew beats his man in the corner and tries to get a shot away from a tight angle, but can't beat Pope at the near post!
90 minFour minutes of stoppage time announced at a tense Liberty Stadium.
90 minSUBSTITUTION: Swansea running down the clock now as Jordan Ayew is withdrawn and Andy King comes on.
90+1 minBOOKING: Andre Ayew goes into the book for a lunge.
4.52pmThat's the final whistle and Swansea's revival under Carlos Carvalhal continues with a 1-0 win over Burnley! It took a while to come but Ki Sung-yeung managed to break the deadlock on 81 minutes to make it three home victories on the bounce in the Premier League for the Swans and move them further away from the relegation zone!
4.55pmThat's all we have time for here at the Liberty Stadium but don't go anywhere as Sports Mole is on hand to bring you the latest match reports, post-match reaction and biggest news from the day's football action! Thank you for following and, until next time, goodbye!