Murphy: Scored only goal
Murphy struck from the left-hand corner of the penalty area after 41 minutes to secure the win.
United were unchanged from the side that won at Bristol Rovers on Saturday but were forced to defend for long spells to keep their League Two hosts at bay.
Cumbrian-born Grant Holt, who supported Carlisle as a youngster, thudded a snapshot volley against the inside of the post after 30 minutes and skipper Ben Davies was just wide with the follow up.
The Shrews had the better of the second half and goalkeeper Ben Williams had to dive full stretch to keep out a shot from Davies who also hit the bar with 10 minutes left.