Leroy Fer was the man to grab the crucial goal, although Lukasz Fabianski more than played his part with a few crucial second half saves.
All in all, a great way to start the first full term under Francesco Guidolin's tutelage.
Swans on top in the first half
Swansea started the brighter of the two sides and were on the front foot from the beginning. In the 8th minute a cross from Mo Barrow found Fernando Llorente in the box and the Spaniard, making his debut for the Swans, could only direct his header into the hands of Burnley 'keeper, Tom Heaton.
Swansea kept possession well with Fer and Jack Cork, in particular, having a good game for the Liberty Stadium side. In the 22nd minute a mistake from Ben Mee allowed Llorente a free shot at goal which was well saved by Heaton in the Clarets' goal.
Swansea continued to press without success, with Burnley's Stephen Ward was fortunate not to be booked for a couple of fouls on the pacey Barrow. Burnley midfielder Dave Jones was booked for a foul in the 42nd as the Lancashire side pressed for the opening goal.
In the 45th minute, Swansea's Jordi Amat had an effort cleared off the Burnley goal-line by Mee as the half drew scoreless to a close.
Better second half
There were no changes from either team at the break and both sides started the half with more intensity. Both clubs upped their pace with Mo Barrow in particular showing the Burnley defence a fast pair of heels. In the 53rd minute Amat was booked for a foul on Andre Gray as Burnley came more into the game.
Gylfi Sigurdsson entered the field to replace Swansea captain, Leon Britton and immediately had a chance for Swansea from a free-kick, just outside the penalty area, which was well saved again by Heaton. From the re-bound, Barrow crossed to Federico Fernandez whose miss on the goal-line could be a contender for the miss of the season.
Fer gets the winning goal
Swansea kept pressing the Burnley goal and in the 82nd minute Fer tapped the ball across the line for the opening goal. Swansea substitute Jefferson Montero beat his man on the by-line and crossed left-footed into the penalty area. The ball was met by a stooping header from Llorente which was well saved by Heaton, only for Fer to be on hand from the rebound and score.
Burnley fought hard to get the equaliser and Gray was unlucky not to score and as the game went into injury time Fabianski made a brilliant save to deny Dean Marney. But the Welsh side held on for a deserved win and the three points, which will surely boost everyone's confidence around the club.