Last season's top scorer Wilfried Bony - who had played for the Ivory Coast on Wedneday - dropped to the substitutes bench, with Bafe Gomis making his first league start of the season.
The visitors were also boosted by the return of Marvin Emnes, who was named on the bench after recovering from a knee injury sustained in pre-season.
Chelsea, meanwhile, made two changes from the side that hit six past Everton as Ramires and Willian made way for Andre Schurrle and Oscar.
SUBS NOT USED: Petr Cech, Filipe Luis, Kurt Zouma, Willian.
SWANS:Lukasz Fabianski, Angel Rangel, Jordi Amat (Federico Fernandez, ht), Ashley Williams (capt), Neil Taylor, Ki Sung-Yueng, Jonjo Shelvey, Wayne Routledge (Jefferson Montero, 66), Nathan Dyer, Gylfi Sigurdsson, Bafetimbi Gomis (Wilfried Bony, 76).
SUBS NOT USED: Gerhard Tremmel, Dwight Tiendalli, Tom Carroll, Marvin Emnes.
Following a minute's applause in memory of Lord Attenborough, who sadly passed away in August, the home side got the game underway at a sun basked Stamford Bridge.
And after a bright start from the visitors in which Wayne Routledge narrowly volleyed wide, the Swans broke the deadlock after 12 minutes.
Following neat interchange in the middle of midfield, Ki Sung-Yueng broke forward before releasing Neil Taylor in space and the defender's low cross was turned into his own net by Chelsea skipper John Terry.
The Swans continued their early dominance as Bafetimbi Gomis twice went close, before Routledge curled a superb effort a fraction wide of the left-hand upright.
Chelsea had to wait until 20 minutes in for their first meaningful effort on goal was Terry headed a Cesc Fabregas free kick onto the roof of the net.
Jordi Amat then picked up the first booking of the afternoon after bringing down Eden Hazard just outside the penalty area, but Fabregas could only fire is effort into the Swans wall as the visitors held firm.
Jonjo Shelvey also entered the referee's notebook for a foul on Hazard, while Andre Schurrle saw the resultant free-kick tipped over the bar by Lukasz Fabianski.
And the Polish goalkeeper had to be at his best just three minutes later as he got down low to save well from Oscar.
But just as it looked as though the visitors would head into the break ahead, Chelsea drew level on the stroke of half-time.
From a Fabregas corner from the right, Diego Costa rose highest to plant a firm header past Fabianski from ten yards out.
Half time: Chelsea 1 Swans 1
Both sides made a change at the break, with Ramires replacing Andre Schurrle for the home side while Federico Fernandez was introduced in place of Amat - who had sustained a knee injury - for the Swans.
Chelsea started the second half brightly with Hazard testing Fabianski before the hosts took the lead on 56 minutes.
After a neat interchange on the ]edge of the box between Hazard and Fabregas, the Spaniard's low centre was rifled into the roof of the net by Costa for his second of the afternoon.
The Swans replied well as Shelvey went close from long range before Gomis was sent clear on goal by Routledge but the French striker could only clip his effort over the onrushing goalkeeper and wide of the far post.
That was Routledge's final action of the game as he was replaced by Jefferson Montero with 24 minutes left on the clock.
But it got worse for the visitors just a minute later as Ramires found space on the edge of the box and his low shot was turned home from close range by Costa, as the Spanish international grabbed his first hat-trick in English football.
That would be Costa's final act as he was replaced by Loic Remy for the final quarter of the game, while Wilfried Bony emerged as the Swans' third and final change in place of Bafe Gomis.
Remy nearly made an instant impact as he fired over from 20 yards, but the striker didn't have to wait long before he got his name on the scoresheet.
After terrific footwork from Hazard inside the box, his pass found Oscar and he laid it on a plate for Remy who curled home into the bottom left hand corner with ten minutes to play.
However, the visitors didn't give up and they grabbed a goal back with four minutes remaining as Bony released Shelvey with a fine pass and the midfielder tucked the ball past Courtois.
The home side could have added to their tally in stoppage time as Remy and Salah shot wide, but it turned out to be Chelsea's day as the Swans suffered a first defeat of the season.