The Premier League side made 11 changes from Saturday's loss to Southampton.
Teenager Aaron Connolly opened the scoring for Albion just after half-time, but League Two side Rovers were back in it thanks to Tom Nichols' superb half-volley.
Murray then was presented with an easy finish with two minutes gone in added time to seal victory for the visitors.
He said: "It was a really pleasing evening for us. It was a tough game - Bristol Rovers made it difficult for us - but the application and the effort from the players was really pleasing.
"Sometimes when a team is just coming together on the pitch for the first time it lacks that little bit of cohesion, but overall I'm happy for the players to get minutes and time, and we're happy to go through."