Rovers fell to their first defeat in five games as Kristian Dennis' goal was the difference at Brunton Park.
Rovers started the game with two changes to the starting XI from last weekend's win against promotion rivals Northampton Town. Glenn Whelan and Sam Nicholson returned to the team in the absence of Antony Evans and Nick Anderton.
Rovers carved open the first chance of the contest just past the five-minute mark. Aaron Collins held the ball up well and played in Harry Anderson on the overlap who crossed the ball into the box where the returning Sam Nicholson wasn't able to direct his volleyed effort on target.
Omari Patrick was played in behind Rovers' defence following a long ball. Expert defending from Connor Taylor prevented Carlisle's No.30 from having any real sight of Belshaw's goal.
Carlisle captain Corey Whelan saw his headed attempt well saved by Belshaw following Joe Riley's pinpoint cross. Moments later Kristian Dennis hit the outside of the post following a shot turned cross. Rovers searched for the opening goal following the brief Carlisle pressure with Elliot Anderson's curling effort being pushed away from danger by Mark Howard in the Carlisle goal.
Former Manchester United man Joe Riley found himself with a chance at goal after getting free at the back post but was only able to place his shot outside of the post and behind for a goal kick.
Brilliant tenacity from Elliot Anderson created a shooting opportunity for the youngster, but his powerful shot from close range was parried well by Howard.
Rovers started the second half on the front foot. Finley's quick shot hit with the outside of his boot was tipped over the bar by the fingertips of Howard. The Gas had penalty appeals waved away moments after by referee Andy Haines after Finley went to the ground.
Luca Hoole reacted brilliantly to a low Carlisle cross into the box as the home side pushed for the opening goal.
Belshaw showed just why the club rewarded him with a new contract this week following a loose touch from James Connolly which allowed Kristian Dennis to have a clear shot at goal. The Rovers No.1 was down well to smother the shot from the Carlisle forward.
The Gas rolled the dice in the 65th minute. Glenn Whelan and Sam Nicholson making way for Harvey Saunders and Leon Clarke. The deadlock was broken in the 71st minute. Kristian Dennis bundled the ball home following pinball in the Rovers area.
Rovers came close to an equaliser straight away. A corner found the head of James Connolly whose effort was cleared off the line by the defending Carlisle players.
Jordan Gibson had the chance to put the home side two to the good following a long throw-in but blasted his powerful attempt well over Belshaw's goal. Luke Thomas entered the fray for Paul Coutts as the game drew to a close. Sam Finley taking the captain's armband from the departing midfielder.
An additional six added minutes were shown by the fourth official as Rovers pushed for the leveller. Elliot Anderson fired in a low cross which was begging to be tapped in but no Rovers players were able to get to the ball as it trickled out of play for a goal kick.
Carlisle were able to cling on to all three points as Rovers faced the long trip back to BS7 empty-handed.
Next up for the Gas is the visit of Mark Hughes' Bradford City to The Mem on Saturday, 2nd April (3pm KO).
CARLISLE: Howard, Armer, Simeu, Riley, Dickenson, Mellish, Feeney, Whelan, Gibson, Patrick, Dennis.
Subs: Norman, Mellor, Devine, McDonald, Alessandra, Sho-Silva, Omotoye.
ROVERS: Belshaw, H Anderson, Taylor, Connolly, Hoole, Whelan, Coutts, Finley, Nicholson, Collins, E Anderson.
Subs: Ward, Harries, L Clarke, T Clarke, Thomas, Saunders, Loft.
Referee - Andy Haines
Attendance - 5,884 (637 away)