Mahrez broke the deadlock with a precision free-kick from the edge of the box that left Chelsea goalkeeper Edouard Mendy rooted to the spot.
And when Mendy denied the Algerian winger five minutes later, Julian Alvarez was in the right spot to turn in the rebound.
Stefan Ortega Moreno, making his third start since joining the Club in the summer, also impressed with a string of fine saves to keep a clean sheet.
WHAT HAPPENED
City had to give everything to secure their dramatic late victory against Fulham on Saturday but they appeared to show no after-effects when they started at a high tempo.
Alvarez, starting his fifth successive match, showed his desire inside the opening two minutes when he closed down a Mendy clearance and the Frenchman was relieved to see the ball deflected wide.
We pushed the visitors back with full-backs Sergio Gomez and Rico Lewis getting forward in support, firing in dangerous crosses that just evaded a finishing touch.
And when Ilkay Gundogan worked an opportunity for Alvarez, his shot on the turn was diverted just wide by Kalidou Koulibaly’s big toe.
But, under new boss Graham Potter, Chelsea offered plenty of threat on the break and Ortega was called upon to make three big saves in the first half, twice denying Christian Pulisic to keep the game goalless.
CARABAO CUP FOURTH ROUND DETAILS
First, he steered the American’s close-range effort wide with his feet when he was diving the wrong way before tipping another curling effort around the post.
His best save came shortly before half-time when Lewis Hall burst through and Ortega spread himself to keep out his low shot.
Pep Guardiola introduced Bernardo Silva and Kalvin Phillips shortly after the restart, the latter playing his first minutes since mid-September after having surgery on a shoulder injury.
And City were menacing after the break with Jack Grealish, who had a dipping effort kept out by Mendy in the first half, twice denied by even better saves.
But the Chelsea keeper had no chance with Mahrez’s 53rd-minute free-kick.
The winger won the opportunity himself with a jinking run that was eventually halted when he was scythed down by Trevoh Chalobah.
And he picked himself up to curl his effort over the ball as Mendy could only stand and watch.
Five minutes later, City added a second with a crisply worked move. Alvarez superb cross-field pass picked out Mahrez, who cut inside and when his low shot was saved, the alert Alvarez was perfectly placed to steer home for his seventh goal of the season.
Raheem Sterling was among a raft of substitutes brought on by the visitors to try to rescue the game and the former City forward was given a warm reception on his first return to the Etihad Stadium since leaving in the summer.
But Ortega was proving to be unbeatable – denying Hall again and even keeping out Pulisic from close range when he was offside.
And when he was finally beaten in the 89th minute by a Nathan Ake deflection after Gomez headed against the post, the effort was ruled out for offside.
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STAR MAN: ORTEGA MORENO
The German keeper made just his second start at the Etihad and was more than comfortable with his feet as we played out from the back and he was unbeatable as the last line of defence with half a dozen strong saves to keep a clean sheet.
