Fernandinho netted the first goal of the new year at the Liberty Stadium, Wilfried Bony replied on two occasions for the hosts, but Yaya Toure and Aleksandar Kolarov wrapped up the points for Manuel Pellegrini's side.
City, who endured their struggles on the road in 2013, got off to the perfect start on a wet and windy afternoon in South Wales, with Jonathan de Guzman inadvertently deflecting a corner into the path of Fernandinho to drill home on 14 minutes.
Swansea grew into the game though - with only their finishing letting them down - and they hauled themselves level in first-half stoppage-time when De Guzman whipped over a cross for Bony to plant a powerful header into the bottom corner - leaving Joe Hart rooted to the spot.
They were, however, unable to keep the back door bolted in the second period and, after the Swans had failed to clear their lines, Toure spun inside the box just before the hour mark to drive past Gerhard Tremmel via the aid of Ashley Williams' studs.
On 66 minutes, City made the game safe as Kolarov was allowed to travel a long way with the ball, cut inside onto his less favoured right foot and drill through Tremmel - with another deflection doing the German goalkeeper few favours.
They were made to sweat a little in the closing stages as Bony lashed in his second of the game from distance, but there was to be no late twist in the tale.
It did not take long for Pellegrini's side to threaten the home goal as Edin Dzeko, Alvaro Negredo and Jesus Navas combined, with the Bosnian firing wide from the edge of box.
Bony smashed a fierce effort just wide at the other end during a lively opening before Swansea suffered the misfortune of losing Pablo Hernandez to injury in just the eighth minute.
The winger, so often hindered by injuries since joining from Valencia, went down clutching his left hamstring and was in tears as he left the field to be replaced by Roland Lamah.
Matters got worse for Swansea when they fell behind in the 14th minute.
Samir Nasri's corner struck De Guzman, and Fernandinho made no mistake in hammering the loose ball beyond Tremmel.
Swansea responded well to going behind, with Jonjo Shelvey firing wide and De Guzman failing to beat Hart after a lovely flowing counter involving Lamah and Bony.
The hosts continued to press and referee Phil Dowd waved away penalty claims against Vincent Kompany when a Williams shot struck the Belgian's arm.
The Swans were given two scares when Negredo and Dzeko failed to capitalise on sloppy play in the home defence, and the failure to take those chances came back to haunt City as Bony, enjoying one of his best days in a Swansea shirt, gave Hart no chance with a header from De Guzman's pinpoint cross.
Flores and Tremmel almost undid all Swansea's good work six minutes into the second half. The pair hesitated dealing with Nasri's lofted pass and Negredo nipped in, only to shoot straight at the keeper.
There was no such reprieve for the hosts in the 58th minute when they failed to clear a Nasri cross and Toure pounced to fire past Tremmel with the aid of a deflection off Williams.
Eight minutes later the clinching goal arrived. Kolarov robbed Wayne Routledge just inside the City half and drove forward unchallenged before cutting inside on to his right foot to beat Tremmel, who ought to have done better.
The excellent Bony worked Hart at the other end before the keeper got a stroke of luck after he came for, and got nowhere near, a De Guzman corner which Williams could only prod wide of an empty net.
Bony pulled one back in stoppage time with a superb low drive, but it was not enough for Swansea.