Two James Berrett penalties and second-half goals by Craig Curran and debutant Liam Cooper sank the visitors, who missed a host of early chances before Berrett opened the scoring on the stroke of half-time. Rovers also played most of the second half with 10 men after Danny Coles received a straight red card.
They could have been well ahead by the interval but for a combination of bad luck and some dreadful finishing. Coles rattled the Carlisle bar with a header and leading scorer Will Hoskins wasted two clear-cut opportunities.
Instead the Cumbrians went ahead in the 44th minute when Gary Sawyer handled in a crowded penalty area and Berrett coolly tucked away the spot-kick.
The second half started badly for Rovers with Coles' 51st-minute dismissal for a challenge on Francois Zoko and Carlisle doubled their lead six minutes later when Curran latched onto Berrett's ball through the middle and neatly slid his shot home.
On-loan Hull defender Cooper struck in the 73rd minute when he rose at the far post to head home a Berrett corner. Berrett scored his second penalty of the game in the 81st minute, blasting home after Sawyer had tripped Ben Marshall in the box.