Michael Laudrup's side gave a good account of itself at the Etihad, but the home side scraped three points despite a rather unconvincing performance.
The first half was a rather drab affair until a late flurry from Swansea, with Michu going close on a couple of occasions, but the scores were level at the break.
A much more potent start to the second half saw City make the breakthrough with 60 minutes on the clock as Carlos Tevez found the net from distance.
Roberto Mancini made three changes from the side which won at West Brom last weekend with Joleon Lescott and Mario Balotelli being replaced by Matija Nastasic and Sergio Aguero, while Aleksandar Kolarov came in for the suspended James Milner.
Swansea also picked up a win last time out in the Premier League and on the back of those three points against Wigan, manager Michael Laudrup opted to name an unchanged side.
The first quarter of the match was somewhat of a non-event, but a deep free kick delivered by Jonathan De Guzman around the 15-minute mark had Joe Hart flapping like a wounded duck. Luckily for the England international the ball went harmlessly out of play.
A Carlos Tevez shot around the 40th minute sparked some life into the match, and Laudrup’s side immediately sprang a marvelous counterattack that ended with Vincent Kompany booting the ball clear ahead of the lurking Michu, after good work down the right from Pablo Hernandez.
A matter of seconds later Leon Britton threaded an inch-perfect through ball between several defenders for Michu, and after getting clear of the City back line the Spaniard looked destined to score, but Hart rushed out and blocked his shot, keeping the score at 0-0 over the break.
City began the second half with purpose as Aguero spun away from De Guzman and then curled an effort towards the top far corner, but it whistled just wide as the match started to liven up.
A couple of minutes later, halftime substitute Mario Balotelli almost got on the end of Samir Nasri’s cross, but Ashley Williams put it behind. The Italian was soon involved again as it appeared he was brought down in the area by Williams only for the referee to wave play on.
However, the reward for City’s good start to the second half came on the hour mark as Gael Clichy burst in towards the center before laying the ball off to Tevez, who fired into the bottom corner from some 30-yards.
Despite being behind, Swansea was not giving up and could have won a penalty with 20 minutes to go as a Michu header appeared to hit Kompany on the arm, while a long-range De Guzman strike narrowly missed the top corner four minutes later.
Twelve minutes of stoppage time were given due to rather nasty looking injuries to Vorm and Micah Richards, but despite plenty of possession in that time, Swansea was unable to to conjure anything up and City held on for three points to go second pending United's match at Chelsea on Sunday.
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