The Swans went down to their first defeat in four games as Manchester City came from behind to win at the Etihad Stadium.
Wilfried Bony had given the Swans the perfect start with a neat finish after nine minutes - the club's first goal at the Etihad.
But Manuel Pellegrini's side were level ten minutes later as Stevan Jovetic directed Jesus Navas' cross into the net.
Fabianski then made a superb save to deny Gael Clichy, with his effort being tipped onto the post, as the Swans soaked up the pressure to go into the break on level terms.
Manchester City took the lead just past the hour as Yaya Toure latched onto Fernandinho's pass to slide home past Fabianski.
The Swans almost grabbed a point, however, as sub Bafe Gomis poked wide while fellow sub Jonjo Shelvey's free-kick was deflected just wide of the goal.
This defeat left the Swans on 18 points from 12 games.
The Swans made one change to the side that beat Arsenal a fortnight ago as Nathan Dyer returned from injury in place of Marvin Emnes.
Tom Carroll kept his place in midfield with Jonjo Shelvey starting on the bench after serving a one-match ban.
Meanwhile, Modou Barrow retained his place in the squad after his starring role in the win over the Gunners with Bafe Gomis - who scored the winner in that game - also featuring on the bench.
MAN CITY: Joe Hart, Pablo Zabaleta, Vincent Kompany (capt), Martin Demichelis, Gael Clichy, Fernandinho (Fernando, 88), Jesus Navas, Yaya Toure, Samir Nasri (James Milner, 79), Stevan Jovetic (Frank Lampard, 70), Sergio Aguero.
SUBS: Willy Caballero, Bacary Sagna, Eliaquim Mangala, Jose Angel Pozo.
SWANS: Lukasz Fabianski, Angel Rangel, Kyle Bartley, Ashley Willams (capt), Neil Taylor, Ki Sung-Yueng, Tom Carroll (Jonjo Shelvey, 67), Gylfi Sigurdsson (Bafetimbi Gomis, 79), Nathan Dyer (Modou Barrow, 77), Jefferson Montero, Wilfried Bony.
SUBS: Gerhard Tremmel, Dwight Tiendalli, Jordi Amat, Leon Britton.
After holding off the early threats from the hosts, the Swans stunned the Etihad by taking the lead after just nine minutes.
Nathan Dyer's perfect flicked pass over the top of the Manchester City defence was chested down by Wilfried Bony, and the Ivorian calmly volleyed home past the onrushing Joe Hart from ten yards out.
But the lead didn't last too long, though, as Manuel Pellegrini's side levelled ten minutes later as Jesus Navas' cross was coolly directed past Lukasz Fabianski by Stevan Jovetic.
The hosts were dominating possession and territory with the Swans having to soak up intense pressure.
With the game past the half-hour mark, Fabianski had to be alert as he rushed out to snuff out Sergio Aguero's low effort after a Samir Nasri cross.
And the Polish international came to the rescue again soon after as he pushed Gael Clichy's thunderous effort onto the post.
Then, just before the break, Fabianski had to twice deal with Jovetic efforts as the Swans reached the break on level terms.
HALF-TIME: Manchester City 1 Swans 1
The Swans started the second half on the offensive, and a well-worked move involving Bony and Jefferson Montero saw the Ecuadorian denied by the onrushing Hart.
But it was the home side who again had the best of possession and Kyle Bartley had to be quick to clear the danger after Fabianski had parried Yaya Toure's effort.
But Toure was not to be denied as he ran onto Fernandinho's sublime pass before slotting past Fabianski after 62 minutes.
Garry Monk made a change five minutes later as Jonjo Shelvey was introduced in place of Tom Carroll to give more attacking options.
Manchester City were pushing for a third, and Pablo Zabaleta drove over Fabianski's crossbar following Toure's inside pass before the Swans made their finals change with Modou Barrow and Bafetimbi Gomis coming on for Nathan Dyer and Gylfi Sigurdsson.
Barrow made an instant impression as his surging run and cross found Bony, and when his effort spun up to Shelvey, the midfielder's attempted volley flew into the Swans supporters behind the goal.
Moments after the hosts almost sealed victory as Aguero's pass sent in Navas, but again Fabianski came racing out to keep his effort out.
The keeper again showed a safe pair of hands to collect Toure's low drive before, at the other end, sub Gomis had a glorious effort to equalise as he latched onto Bony's through ball. But the French striker's effort bounced well wide.
And with the game inside the four minutes of added on time, Shelvey's deflected free-kick dropped just wide of the top corner and out for a corner.
But that was the end of the action as the Swans bravely went down to defeat.