Daniele Dichio's scored for the home side with a header half way through the second half open the scoring for West Brom.
Cheltenham had chances, but couldn't get a goal back.
After the game Steve Cotterill said:“The players have done nothing differently today to what they have done in all the league and cup games this season.
“They don’t change, they are a terrific bunch with feet firmly on the ground. They enjoyed themselves today and I think deservedly so.
“They didn’t let anybody down and I’m delighted with them. I thought they were great.
“Going into the dressing room they were disappointed but they have nothing to hold their heads down about.
“They can hold their heads high. They have come to a very difficult game and equipped themselves in a good manner.”
Cotterill believed that Cheltenham dealt with West Brom well until Daniele Dichio scored the only goal of the game after 64 minutes. He also reflected on the one really good chance that his team had.
“At the beginning of the second half West Brom put us under a bit of pressure but they didn’t really have a clear shot at goal.
“You can’t expect to come here and not expect to come under pressure. That was always going to happen at some stage but we dealt with it quite well.
“The boys did not freeze and we played some good stuff at times. I thought Jamie Victory’s header had gone in. I think it was Chambers who cleared it off the line. I didn’t see where he came from and I wish he hadn’t. It was a bit of a sickener when it was cleared but that’s how it is."
Cotterill thanked the 5,500 travelling fans who contributed to a marvellous atmosphere at The Hawthorns, and wished his opposite number Gary Megson the best of luck in the next round.
“There was great backing from our supporters but we had 150 at Darlington on Tuesday night and they were great too.
“It’s been a good ride for everyone involved with the club and I’ve just told the players that in about 10 months time they can prepare because this is what it’s all about.
“I would like to wish West Brom all the best of luck in the next round and for the rest of the season. From the minute the draw was made they have been fantastic to us so good luck to them.”