Ellis Chapman came the closest to breaking the deadlock in the first half, as his long-range strike bounced back off the frame of the Rovers goal.
Alfie May grabbed the tie’s first goal as the game approached the hour, finishing from close range after being found by Wright’s delivery into the Rovers area.
Ahead of the tie, Rovers made five changes to face the Robins.
Connor Taylor and James Belshaw both were named in the starting XI for their Gas debuts – with Alfie Kilgour, Jack Baldwin and Ben Liddle also listed as changes to the side that faced Mansfield on the opening day.
Early chances were at a premium, with only Callum Wright’s speculative volley from outside the area the only opportunity of note inside the opening ten minutes.
Rovers enjoyed the majority share of possession, with the mainstay of the play focused in the middle of the pitch. A short corner routine involving Thomas and Westbrooke saw the latter drill a low cross into the congested area that narrowly ran out of the reach of Harvey Saunders at the near post.
On the half-hour mark, a long-range strike from Ellis Chapman kept low and skidded across the plush Mem surface, bouncing back off the inside the Rovers upright, in what a fortunate left off for the Gas.
As the first half progressed, Rovers had appeals for a penalty turned away when Saunders and Tozer collided inside the Cheltenham area. Shortly after, Anderson aimed a curling effort towards that target that came close to forcing Flinders into action.
With the deadlock in place at the interval, Rovers had a disallowed goal early on in the second half when Harry Anderson was judged to have been offside when latching onto a useful through ball and lifting a finish beyond Flinders.
The game’s first goal came just on the turn of the hour. A searching ball over the top from Callum Wright found the run of Alfie May, who kept his balance to chip a composed finish over the onrushing Belshaw.
Cheltenham made a double change after going in front, introducing Kyle Vassell for his debut and also Liam Sercombe, who received a warm welcome on his return to BS7.
Rovers made a double change of their own – with a debut for the Academy’s Jamarni Langlais and Sion Spence coming off the bench.
It was the Cheltenham substitute who had an instant impact, as Kyle Vassell was in the right position at the far-post to react to the ball pinballing around the Rovers penalty to convert home from close-range.
Rovers’ third and final change saw Ryan Jones called into the fold, with the youngster also making his EFL debut.
Rovers host Stevenage at Mem in Sky Bet League Two on Saturday.
Rovers: James Belshaw, Connor Taylor, Alfie Kilgour, Nick Anderton, Jack Baldwin, Luke Thomas (77′ Ryan Jones), Zain Westbrooke (75′ Sion Spence), Ben Liddle (75′ Jamarni Langlais), Josh Grant, Harvey Saunders,
Subs: Jed Ward, Cian Harries, Luca Hoole, Joe Budd.
Cheltenham Town: Scott Finders, Ben Tozer (c), Alfie May (65′ Kyle Vassell), Callum Wright, Sean Long, George Lloyd, Grant Horton, Taylor Perry (65′ Liam Sercombe), Ellis Chapman (79′ Chris Hussey), William Boyle, Lewis Freestone.
Subs: Owen Evans, Charlie Raglan, Elliot Bonds, Callum Ebanks.