The hosts started brightly and soon began to dominate possession, with their first chance coming after just three minutes when Jason Shackell was well placed to steer Wilfried Bony’s goal-bound header off the line.
Burnley did threaten in spells as Ings fired over before Lukasz Fabianski cut out his low cross looking for Lukas Jutkiewicz, while Tom Heaton denied Gylfi Sigurdsson from inside the area with a comfortable save following Wayne Routledge’s low cross.
Swansea took the lead on the 23rd minute and the lively Sigurdsson, making his second home debut for the club following his summer switch from Tottenham, cut apart the Burnley defence as Nathan Dyer raced onto his through ball to fire beyond Heaton.
Burnley were the better side for the majority of the second half with Jutkiewicz, Scott Arfield and Ings all threatening in the early stages. Jutkiewicz spurned their best chance when he headed Dean Marney's deep cross over the bar after losing his marker.
David Jones then burst into the Swansea box and Fabianski was forced into a fine save to tip his low strike wide after some smart play from Ings before subsitute Ashley Barnes fired over in stoppage time, but Garry Monk's side held on to make it two wins from their opening two games.
The hosts were unchanged following their shock victory at Old Trafford, with midweek signings Federico Fernandez and Tom Carroll introduced to the crowd ahead of kick-off but not included in the matchday squad.
Burnley were also unchanged from their 3-1 home defeat to Chelsea as Sean Dyche's men looked to get their first points on the board following promotion from the Championship.
Swansea went close to taking a third-minute lead when Wayne Routledge's cross found Bony 12 yards out. The Ivorian's header seemed destined for the far corner of the net with Heaton sprawling, only for Shackell's precious interception to keep him out.
Burnley were not shy in coming forward and Ings was lively as he played off summer signing Jutkiewicz in attack.
The England Under-21s striker volleyed over from just inside the box after Jonjo Shelvey had been dispossessed before another quick attack saw his cutback grasped by the grateful Fabianski with Jutkiewicz lurking.
Shelvey soon forgot about his earlier error, though, as he got Swansea moving with a raking 40-yard pass which released Routledge down the right.
Routledge sped past Kieran Trippier to deliver the perfect ball for Sigurdsson but the Icelandic midfielder's shot lacked power and Tom Heaton held on.
But Heaton was beaten five minutes later as Ben Mee failed to intercept Sigurdsson's ball and Dyer advanced to beat the former Manchester United trainee inside his near post with a low drive.
The goal settled Swansea down and Bony began to flex his considerable muscles but Burnley were well-organised and refused to yield too much space.
Sigurdsson volleyed over from the edge of the area following a poor clearance from Michael Duff, who was making his 500th career appearance, and Bony's weak shot was comfortably held by Heaton before the break.
Bony had another chance straight after the restart when Dean Marney's loose header fell into his path but was unable to capitalise with a horrible mis-kick as he bore down on goal.
But Burnley were still very much in the game and Jutkiewicz felt he should have had a penalty after more good work by Ings, although television replays showed Ashley Williams made contact with the ball.
Scott Arfield and Marney both saw wayward efforts fly wide and Ings' drive was deflected wide as the home crowd began to get restless.
And Burnley should have been level when Marney's 64th-minute cross picked out the unmarked Jutkiewicz but he could only head over Fabianski's bar after ghosting free.
Garry Monk had seen enough and sent on Bafetimbi Gomis and Jefferson Montero for Bony and Dyer - but Burnley were within inches of equalising when Ings found David Jones and his shot was saved by Fabianski, with the ball just evading Jutkiewicz, who slid in at the far post.
Swansea almost released the pressure valve when Montero's cross was punched out by Heaton and Ki Sung-yeung's rising drive just cleared the crossbar.
Burnley continued to press in the final minutes and Barnes, a January signing from Brighton, lashed over the bar as Swansea edged to the three points.
Charlie Nicholas on Swansea v Burnley
I thought Burnley deserved something from this, but they walk away with nothing. In the first half Swansea bossed it and I don’t know what Sean Dyche said to them, but in the second half they upped the tempo, drove forward and pinned Swansea in. They were the better team in the second half, but didn’t make a bundle of chances. If it had gone 1-1 they might have gone on and nicked it.
It was a cracking game to watch and both teams were very good, but I would have to say that Burnley will be disappointed not to get anything from it. I like a lot of Swansea’s signings. I don’t see them being anywhere close to a scrap. What a sensational start it is for them.