City scored three times in the final 10 minutes to overcome the League Two side, who provided a scare when they took a surprise lead on the hour mark.
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Seventy-two league places separated the two teams, but having matched Pep Guardiola’s side in a goalless first half, the Robins went ahead when Alfie May converted from Ben Tozer’s long throw in.
Phil Foden, who impressed throughout, netted a composed volley from Joao Cancelo’s cross to equalise nine minutes from time and Gabriel Jesus completed the turnaround moments later when he collected Fernandinho’s superb pass and fired emphatically past the Cheltenham goalkeeper Josh Griffiths.
Ferran Torres tapped in Ilkay Gundogan’s cutback from close range to wrap up the victory with the last kick of the game and set up a fifth-round tie with Swansea City.
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