Graham Coughlan admitted an element of relief as his Bristol Rovers side picked up their first win and goals of the season at the third attempt.
Tyler Smith fired the hosts ahead on 38 minutes after Cheltenham goalkeeper Scott Flinders parried a Kyle Bennett volley and doubled the advantage from Ollie Clarke’s 49th minute pass before teeing up Jonson Clarke-Harris for a 67th minute tap-in.
Rovers have been drawn at home to Brighton in the second round and Coughlan said: “It’s great that a Premier League club are coming here and it will have the club buzzing.
“But we need to focus on getting our League campaign going before that. I am relieved to have got a win and was pleased to see us scoring some goals.
“I wasn’t happy after the opening day defeat at Blackpool or following Saturday’s goalless draw with Wycombe. Tonight there was progress, but I won’t be satisfied until we are producing that sort of performance on a consistent basis.
“I was only intending to play Jonson Clarke-Harris for 45 minutes because of the injury that disrupted his pre-season. It is testament to the way he handled his general fitness that he was on for as long as he was and got his goal.
“Tyler Smith looked very sharp and I have a number of selection problems ahead of the weekend, which is what I wanted.”
Cheltenham boss Michael Duff had no complaints.
“The better side won,” he said. “I thought we started well and I wasn’t too worried when we went in a goal down at half-time because I felt we were still in the game.
“But I was disappointed with the way we played after their second goal. We had just missed a good chance and had begun the second half well.
“From 2-0 down, it was men against boys. Rovers passed the ball better than us and I was disappointed with some of my more experienced players because the young lads did well.
“We are a better team now than when I first came in, but tonight some of the players were below their best.”