Nathan Dyer scored the game's only goal, with visiting keeper Tom Heaton failing to keep out his low shot.
Burnley came close to equalising through Lukas Jutkiewicz and David Jones in the second half.
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But Swansea withstood the pressure and, following last Saturday's first league win at Manchester United, are level on points with top-of-the-table Chelsea.
Defeat for Burnley, meanwhile, leaves them still searching for a first Premier League point of the campaign, although their battling display will have been encouraging for manager Sean Dyche.
Swansea boss Garry Monk named an unchanged team from the opening-day triumph at Old Trafford, with new signings Federico Fernandez and Tom Carroll left out of the squad.
Burnley also stuck with the same side which lost 3-1 to Chelsea on Monday, although Stephen Ward was named on the bench after joining from Wolves.
The opening defeat by Chelsea was a chastening reintroduction to Premier League football for the Clarets, who were overwhelmed in midfield, and they were similarly outnumbered by Swansea.
The hosts stretched their opponents at every opportunity, with wingers Dyer and Wayne Routledge particularly threatening in the opening exchanges.
Routledge created early chances for Wilfried Bony and Gylfi Sigurdsson, but it was Dyer who opened the scoring.
Latching on to Sigurdsson's through ball, Dyer scampered into the penalty area and shot low into the net despite a touch from Burnley keeper Heaton.
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Swansea were well worth their lead and, apart from a swerving 20-yard shot from Burnley striker Danny Ings, Monk's men were not troubled during a comfortable first half.
They faced a reinvigorated Burnley side after the restart, however, and the visitors were close to equalising when Jutkiewicz headed narrowly over after an hour.
Jones then had a shot palmed wide by Lukasz Fabianski, while Swansea brought on summer signings Bafetimbi Gomis and Jefferson Montero for Bony and Dyer as they sought to regain the initiative.
Although Gomis and Montero injected pace and urgency into the home attack, it was the Swansea defenders who were most influential in the closing moments.
With captain and centre-back Ashley Williams a steadying presence on his 30th birthday, the Swans repelled a flurry of late Burnley attacks to cling on to a second successive win and consign the Clarets to another defeat.
Swansea manager Garry Monk: "It's a fantastic start and great to have six points.
"It's the best start we could have had, but we know we are only two games from being on your back and getting dragged down.
"We have to build on what we've done and improve on mistakes."
Burnley manager Sean Dyche: "We showed them too much respect in the first half and that can happen for a team that's just been promoted.
"I want them to play with freedom, quality and energy and I thought we did that in the second half.
"I want us to be brave, but not in a naive way. We want to play on the front foot, create chances and win games."
Swansea City made it two wins from two Premier League games as they held on to beat promoted Burnley.