In a game of precious few chances, it was Cesc Fabregas’ superb fourth-minute strike that proved decisive – his 50th in the Premier League.
Andre Ayew curled wide for the hosts late on, but they rarely looked like finding a leveller against opponents who comfortably kept them at arm’s length throughout.
The win lifts Antonio Conte’s side just two points behind fourth-placed Tottenham, who face Watford at Wembley on Monday.
Swansea, meanwhile, are just one point ahead of 18th-placed Southampton following Mark Hughes’ side’s 2-1 win over Bournemouth earlier in the day.
Carlos Carvalhal’s men now have three games to preserve their top-flight status, including home clashes with the Saints and Stoke City, in what promises to be a nail-biting climax to the season.
Swansea barely had a chance to catch their breath by the time Chelsea had raced into an early lead.
Eden Hazard showed characteristically quick feet to find space on the edge of the penalty area before teeing up Fabregas, who whipped a first-time effort with his left foot into Lukasz Fabianski’s top corner.
Swansea were successful in disrupting Chelsea’s flow at the start of the second period but, much like in the first-half, they rarely looked like finding an equaliser.
Tiemoue Bakayoko dragged a shot wide from distance as Chelsea looked to double their advantage and put the game to bed, while Emerson Palmieri’s low drive was kept out by Fabianski after the Brazilian had produced a dazzling run from just inside the Swans’ half.
Moses then somehow contrived to prod wide from all of three yards after Fabregas’ shot had careered towards him before Ayew missed the hosts’ best chance.
The Ghanaian picked the ball up just inside the area, but his left-foot shot arced agonisingly wide of Courtois’ post.
Tom Carroll flashed a shot wide and Wayne Routledge fired straight at Courtois in the closing stages as Swansea ultimately fell short of finding a priceless equaliser.