But City equalised when Kevin de Bruyne’s shot hit Kurt Zouma and wrongfooted goalkeeper Kepa Arrizabalaga.
Riyad Mahrez put the champions – who move above Chelsea into third place – ahead with a fine finish after cutting in from the right wing.
Both sides could have scored more in a chaotic first half, City going closest when Sergio Aguero hit the crossbar after a poor clearance by Kepa.
There were fewer incidents in the second half although Aguero hobbled off with an injury and had to be replaced by Gabriel Jesus.
Raheem Sterling thought he had scored a late third for City but it was ruled out by the video assistant referee in a very tight offside call.
City sit nine points behind leaders Liverpool, who beat them 3-1 in the last game before the international break.
