Chelsea trailed to Gylfi Sigurdsson’s strike following an increasingly sloppy defensive display at the Liberty Stadium.
The Blues posed the greater threat during a scrappy opening 25 minutes, with Oscar and Ruben Loftus-Cheek both going close.
But after Sigurdsson scored with a volley from the edge of the area to put Swansea ahead, Chelsea looked disorganised at the back and could have been further behind at the break.
Blues goalkeeper Asmir Begovic, who making his first Premier League start since November, made a fine save from Sigurdsson with the score at 0-0.
But, like all of his back four, Begovic was far from convincing and only some wayward finishing and poor decision making stopped Swansea from adding to their lead.
Centre-backs Matt Miazga and Branislav Ivanovic had several nervy moments while full-backs Cesar Azpilicueta and Baba Rahman looked nervy against the home side’s pace out wide – particularly when facing Jefferson Montero.
Before the Swans went ahead, Chelsea had just started to get on top, with Oscar close to opening the scoring.
The Brazilian’s effort went just over following a corner, while Ruben-Loftus Cheek had a shot easily saved and was then denied a tap-in by some last-ditch defending.
Oscar was also involved in Chelsea’s best chance in first-half injury-time but he took too long after being put through by Cesc Fabregas and Pato shot wide as he followed up.
Gary Cahill failed his late fitness test, meaning Miazga is partnered in central defence by Branislav Ivanovic, who captains the team in the absence of John Terry.
Oscar also came in, for Kenedy, while youngsters Ola Aina, Charlie Colkett and Kasey Palmer are on the bench.
Subs: Nordfeldt, Naughton, Amat, Ki, Barrow, Routledge, Gomis.
Chelsea: Begovic; Azpilicueta, Miazga, Ivanovic, Baba Rahman; Mikel, Fabregas; Pedro, Loftus-Cheek, Oscar; Pato.
Subs: Courtois, Aina, Kenedy, Colkett, Palmer, Traore, Falcao.