It was a far-from-vintage performance from City, who faced a well-drilled Bournemouth defence, but Mahrez slotted home in the 55th minute to hand the visitors the win.
Pep Guardiola was dealt a blow, though, as Kevin De Bruyne was forced to come off just before the break after picking up another injury, with the manager saying after the game that it was “muscular”.
City go two points clear of Liverpool – who play Everton on Sunday, at the top of the Premier League while Bournemouth remain in 12th and winless in five league outings.
City dominated possession in the first half, but did not create many clear cut chances. They had their first real sight of goal in the 19th minute as Bernardo Silva slotted the ball to De Bruyne down the right-hand side. He then cut the ball back for David Silva in the area, but he could only pull his shot wide.
Just after the half an hour mark, Nicolas Otamendi could have put City ahead in style. De Bruyne delivered a corner and with Otamendi having peeled away from his marker, he attempted an audacious bicycle kick but he sent it just over the crossbar.
City were dealt a blow on the stroke of half-time as De Bruyne went off injured. As he made a pass, he appeared to pull a muscle and signalled that he needed to be replaced. He looked disappointed as he made his way off, having already missed two spells this season with injuries, and was replaced by Mahrez.
Bournemouth made three changes with Charlie Daniels, David Brooks and Jack Simpson coming into the XI. Man City also made four changes from their midweek game, bringing in Kyle Walker, John Stones, Bernardo Silva and Raheem Sterling.
And it was the Algeria international who finally put Man City ahead in the 55th minute. After some good inter-link play between David Silva and Oleksandr Zinchenko, the former picked out a waiting Mahrez on the right-hand side. He then hit home on the angle, sending the ball into the bottom, far corner.
The remainder of the game continued much like the first half – plenty of Man City possession but few clear cut chances. However, the visitors did go close twice in the final 10 minutes.
In the 81st minute, Sergio Aguero floated a dangerous ball into the area that, although appearing to be intended for Mahrez, pinged onto the crossbar before going wide with Boruc getting a fingertip onto it. The Bournemouth goalkeeper made a wonderful save not long after as Mahrez headed in his direction from point-blank range, but Boruc pushed the effort away.