The Clarets side made it six successive league victories as they stepped up their chase for promotion back to the Premier League after last season's relegation.
Ian Maatsen fired Vincent Kompany's side into a two-goal lead with a pair of fine goals in the space of 10 first-half minutes at the Liberty Stadium.
Ollie Cooper got one back for Swansea in the 27th minute, but Burnley were able to hold for another three points.
Burnley are five points clear at the top after second-placed Sheffield United needed a stoppage-time goal from John Egan to salvage a 1-1 draw at QPR.
The hosts led through a deflected strike from Ilias Chair in the 11th minute.
Egan's shot through a crowded area deflected home in the final seconds to spare the Blades from a damaging defeat.
Watford moved up to fourth place as Vakoun Bayo grabbed a late goal to defeat managerless Norwich 1-0 at Carrow Road.
The Hornets recovered from successive defeats as Bayo finished off a swift counter-attack with three minutes left.
Middlesbrough are only a point behind Watford in fifth place after securing a sixth win in seven games, beating Birmingham 3-1 at St Andrew's.
Matt Crooks was unmarked to head in Jonny Howson's cross in the 57th minute and seconds later he netted again after a mistake from George Friend.
Although Tahith Chong pulled one back, Crooks then played in Chuba Akpom to make sure of the points for Michael Carrick's side.
West Bromwich Albion moved to within one point of the play-off places as Daryl Dike's 60th-minute goal earned a 1-0 win against Reading.
Bottom of the table Wigan suffered a third 4-1 defeat in the space of seven days as Hull moved away from danger with an emphatic victory at the DW Stadium.
Ched Evans headed the stoppage-time winner as Preston snatched a 1-0 victory at Stoke.