Ki Sung-yueng gave his new manager even more to smile about after hitting the only goal of the day at the Liberty Stadium on Saturday afternoon.
Patience was the name of the game as fans had to wait until well into the second half for the first shot on target from either the hosts or visiting Burnley.
Andre Ayew worked his socks off down Burnley's left and displayed a man of the match kind of performance and was instrumental in the goal which came through an 18-yard strike from the unlikely Ki.
Carlos Carvalhal made six changes to the side that put eight past Notts County mid-week, with in from striker Joran Ayew coming in for Tammy Abraham upfront.
Burnley were still without eight of their first team players with James Tarkowski, Phil Bardsley, Steven Defour and Chris Wood all still out, amongst others - Sean Dyche opting for Sam Vokes and Ashley Barnes up front, who together helped snatch a point from Manchester City last week.
Vokes came closest early on in what really was nothing more than a half chance at best. Holding the ball up on the edge of the Swansea box, the striker managed a snapshot but the effort landed just wide of the post.
A few more half chances fell for the away side as the first half neared its midway point with Barnes lashing a shot just over Łukasz Fabiański's crossbar but the deadlock remained as both teams continued to cancel each other out.
The first half increasingly confirmed the lowkey uneventfulness of the match - by the 30 minute mark neither team had registered a shot on target, though Swansea did start to test Burnley's backline a little more with some decent crosses in from both sides.
Around the 36-minute mark Ki found Sam Clucas in a great position just outside the Burnley box - the striker was given the space to turn and shoot yet failed to force the keeper into any sort of save with the ball clearing the crossbar by at least a yard or two.
Swansea found a nice spell inside the Burnley half toward the final minutes of the first half with Jordan Ayew showing off his dribbling ability down the right, though yet again the chance came to nothing with Burnley's defence showing why they are seventh in the table.
The first half ended with both defences doing their job well - no real chances had been created at either end. Swansea showed great desire to make stuff happen going forward under their new manager, pleasing the home crowd, though Burnley had enough to match their efforts.
The first shot on target came through Jóhann Guðmundsson in the 55th minute, as the midfielder was found through a long diagonal out to the right, though Fabiański was able to tip the shot over the bar rather comfortably.
Swansea didn't wait long into the second half to make a change with Martin Olsson making way for André Ayew in the 59th minute, who joined his younger brother Jordan in attack for the remainder of the match.
Ben Mee came close to breaking the deadlock through a Burnley set piece with a great ball in from the sidelines; the defender rose high but pressure from the Swansea defence was just enough to force the header just wide.
It didn't take Abraham long to make a mark on proceedings after nodding on a whipped in cross from the right, the pace of the ball shaking the keeper a little, though the ball found its way into the grateful gloves of Pope in the end.
The deadlock was finally broken in the 81s minute though an excellent drive into the centre from Kyle Naughton. Burnley failed to close down the right back and after a nice touch from Andre Ayew, Ki was found with enough space to shoot, just outside the box. The ball found the back of the net and the Liberty Stadium came alive after a painful 81 minutes of silence.
The gaps began to open up as Burnley pushed for the equaliser but the added effort from the away side only allowed Andre Ayew acres of space to run into. The striker displayed his skill in possession once again, twisting around the breathless Burnley defenders but to no avail as his toe-poked shot was saved by the keeper.
The win takes the Swans up to 15th, at least temporarily, while Burnley remain in seventh - without a win in their last 10 Premier L