The League Two side, 72 places below City in the football pyramid, looked set to pull off one of the biggest modern-day FA Cup shocks after taking a second-half lead.
But City, prompted by the inspirational Phil Foden, scored three times in the last ten minutes through Foden, Gabriel Jesus and Ferran Torres to navigate the trickiest of ties at the Jonny-Rocks Stadium.
City were almost in front after just 24 seconds as Riyad Mahrez’s superb pass put Gabriel Jesus in behind the Cheltenham defence but the Brazilian’s attempted lob was tipped away by Robins keeper Josh Griffiths.
The hosts went close themselves immediately with Alfie May inches away from connecting with a near-post cross in a bright opening to the game.
Benjamin Mendy thought he’d broken the deadlock on 12 with a thunderous effort from just inside the box but Cheltenham skipper Ben Tozer somehow managed to head off the line.
Ferran Torres was next to go close, connecting well with a Phil Foden cross but again Griffiths saved well.
There was no hint of frustration in City’s play with Cheltenham well-organised and committed in the challenge.
With so many changes to the starting line-up, it was always going to be a more workmanlike City performance, particularly in the first-half which was held up for five minutes as a mini firework display behind the stand saw the referee briefly suspend the action.
The half ended goalless, with the League Two side holding their own and gradually growing in confidence.
Eight minutes after the restart a fine through-ball from Foden set Gabriel Jesus clear, but he chose to go inside as the keeper approached and rolled his shot against the outside of the post.
It was the best chance of the game, but with one goal in 12 games, Jesus' confidence in front of goal isn't at its highest.
Then, just before the hour-mark, the sting from the hosts as one of Tozer's long throws caused confusion and Alfie May was first to the loose ball, firing high into the roof of the net from a few yards out.
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With half-an-hour to play, there was time to save the blushes.
On 70 minutes Foden again sent a team-mate clear - this time Mahrez - but his show was again well saved by Griffiths.
Then, finally, with the clock ticking into the last 10 minutes, Foden - our best player on the night - followed sub Joao Cancelo's cross into the box before volleying home from close range.
The relief was palpable.
Two minutes later, City were ahead.
Patient play again by a City side who had refused to panic ended with Fernandinho picking out Jesus' clever movement and the Brazilian chested the ball down before spinning and volleying home from eight yards.
It was quite a turnaround.
And it wasn't over, with Ferran Torres grabbing a third with the last kick of the game, turning in Ilkay Gundogan's low cross to perhaps give the scoreline a more comfortable look than it really was!