Grant McCann's side looked as if they were destined to not break the deadlock against opponents that were proving stubborn despite playing most of the second half a man light.
But Jamie Sterry and sub Rob Street notched late on to spark joy in the away end.
The first half was a tight, tense tussle with neither side creating a plethora of chances. The closest Rovers came was a snapshot from Jordan Gibson that was straight at Joe Day, whilst Jordan Thomas fired a fierce effort just wide of Ted Sharman-Lowe's left-hand post in response.
Early in the second half came the turning point when Cheltenham's Tim Dieng was shown a second yellow card for a lunge on Jack Senior. With a man advantage, Rovers looked to turn the screw and Gibson struck the crossbar from range almost immediately.
But Cheltenham were a tough nut to crack. Just as it looked as though spoils would be shared, up popped Sterry with a vicious effort for his first Rovers goal. Street then added gloss with a classy lobbed finish in stoppage time.
Here's our player ratings from Cheltenham:
9. Luke Molyneux 7
Did his best to grab the game by the scruff of the neck and looked Rovers' best output for a goal. Teed up Sterry for the game's vital moment. Photo: Howard Roe/AHPIX LTD
10. Jordan Gibson 6
Fired a couple of attempts towards goal but nothing to overly worry Joe Day. Smacked the bar shortly after the red card. Photo: Howard Roe/AHPIX LTD
11. Billy Sharp 5
A struggle. Too often was isolated up top or balls went straight to the head of his marker. A couple of attempts but neither on target. Photo: Craig Hawkhead/AHPIX LTD
12. Jay McGrath 5
Brought on for Wood in a big boost as we enter the run-in. Had little to trouble him. Photo: Howard Roe/ahpix.com