Roberto Mancini's side put their 3-1 loss at Ajax behind them with a 1-0 success at home to Swansea City.
Defending champions City moved up to second place and closed to within a point of leaders Chelsea, who host third-place Manchester United on Sunday.
It took City 38 minutes to register a shot on target against Swansea in a lacklustre first half at Eastlands.
Swansea looked more likely to take the lead, but their clearest opportunity saw Joe Hart race from his line to save at the feet of the advancing Michu.
Michu drew Hart into action again with a downward header in the 59th minute but two minutes later, Roberto Mancini's side struck.
Gael Clichy was allowed to carry the ball infield before finding Carlos Tevez, and the Argentine sent a shot dipping into the bottom-left corner to end a run of eight games without a goal.
Swansea goalkeeper Michel Vorm had to be stretchered off after appearing to hurt his groin as he tried to reach Tevez's shot, while City defender Micah Richards also had to leave the fray with a leg injury.
The stoppages meant that over 12 minutes of injury time were played -- breaking the Premier League record -- but City managed to keep their slender lead intact.
