City came close to go ahead before halftime through the head of Gabriel Jesus following a smart delivery into the area, but it went wide of the mark.
Jesus was inches away again minutes following halftime when he clipped the upright from a well-weighted through ball by Phil Foden.
Cheltenham stunned the Citizens on 59 minutes after Alfie May capitalised on a touch at the near post from George Lloyd following a long throw-in to beat Zack Steffen.
Foden drew the Citizens level nine minutes before time through a sumptuous volley from the back-post after a cross from Joao Cancelo left the host unstable at the back.
It was two minutes later after Jesus atoned for his earlier misses when he beat the offside trap to bring down a ball over the top before riffling past Cheltenham goalkeeper Joshua Griffiths.
The contest was put to bed at the death after Ferran Torres made use of a pass from Ilkay Gundogan to fire in from point-blank range, as the Citizens ensured fifth round progression.
