The Bosnian forward scored in the 86th minute to hand Roberto Mancini's side a late win following a questionable penalty award to the hosts from referee Mark Halsey
The hosts went ahead in the eighth minute thanks to a Mladen Petric penalty after John Arne Riise was ruled to have been fouled in the box, but Sergio Aguero levelled the scores in the 42nd minute.
The home side carried their hard work into the second period, continually denying City’s players any space or time on the ball, but the champions found a winner in the 86th minute thanks to substitute Edin Dzeko.
Fulham made two changes to the side that beat Wigan 2-1 in their last game, with Bryan Ruiz and Petric replacing Alex Kacaniklic and the injured Dimitar Berbatov.
Meanwhile, Roberto Mancini made eight changes from the side that lost 4-2 to Aston Villa in the League Cup, with only Carlos Tevez, Gareth Barry and Matija Nastasic keeping their places.
Both sides began the game well but it was the hosts who were given an early boost in the eighth minute. Riise was taken down by Pablo Zabaleta and, despite protests from the away side that the foul was outside the box, referee Mark Halsey awarded a penalty.
Petric subsequently stepped up confidently to send Joe Hart the wrong way, giving Fulham a 1-0 lead in just the ninth minute.
The Craven Cottage crowd were given a tremendous lift by that early strike, and despite some intricate passing by the away side, the Fulham defence held firm to stop any space opening up.
The Cottagers then had a glorious chance for a second in the 15th minute, as after a superb through ball found Hugo Rodallega, he played it back to Petric who unfortunately blazed his shot over the bar.
Fulham’s attacking players looked a danger to the fragile City defence, and the away side struggled to break down the obdurate pairing of Chris Baird and Steve Sidwell in the opening 20 minutes.
However, City almost found a way through in the 24th minute, as a superb ball from Tevez found Zabaleta at the back post, and his pass back into the penalty area was only just cleared by a Fulham defender.
The away side then had another superb chance, as Aguero found himself with a free header at the back post from a corner, but was only able to put the ball into the side netting.
City then had a shout for a penalty of their own as Tevez was pulled to the ground by Riise having run in behind, but referee Halsey rejected the away side’s claims.
However, the champions' defenders were still having problems with Fulham’s attacking threat, with Vincent Kompany picking up a yellow card in the 35th minute after fouling Ruiz just outside the penalty area.
From the resulting free-kick, Ruiz managed to pick up a rebound after the ball cannoned off the wall, and saw a shot hit the side-netting, although Hart looked to have it covered.
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In an instant, City took the ball into Fulham’s penalty area, and nearly had an equaliser as the ball trickled close to the goal line, but Mark Schwarzer smothered the ball just in time.
However, the away side finally found the equalising goal in the 42nd minute, as a cross came into the box that was parried by Schwarzer, and Aguero was there to tap home and level the scores.
With the scores level at the start of the second half, it was the Cottagers who had the best early chance in the 49th minute, as a Ruiz free-kick caused havoc and was only just cleared by the City defence.
It took until the 59th minute for either side to create another clear chance, and it was the Premier League champions who did, as Aguero glanced a header wide after a teasing cross from David Silva.
Opportunities became hard to come by from that point, before substitute Mario Balotelli saw a shot from outside the box go just wide of Schwarzer’s post in the 69th minute.
Just three minutes later, Fulham had a chance of their own after some good work from Rodallega led to Ruiz hitting a powerful shot which was saved well by Hart.
However, in the 86th minute, the away side finally had a reward for their relentless pressure, as from a good cross, second-half substitute Dzeko turned superbly and powered the ball into the net to give the champions the lead, which they kept until the final whistle.