Newly crowned World Cup winner Julian Alvarez doubled their lead on the half-hour mark after a VAR review saw the Argentina striker spot a handball from Kai Havertz.
Phil Foden completed a clever team move involving England team-mate Kyle Walker to score City’s third goal in the 38th minute, effectively ending the game as a contest.
The hosts won another penalty late in the second half and although Álvarez was still on the pitch, Mahrez took the ball and slotted past Chelsea goalkeeper Kepa Arrizabalaga in the 85th minute.
City will play either Premier League leaders Arsenal or League One side Oxford United in the fourth round, which meet on Monday.
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