Bury midfielder Steven Schumacher sparked a second-half fightback and the Shakers were unlucky not to grab a share of the points.
Carlisle made a dream start with two early goals. The first of them came in their first attack in the second minute when Brad Potts cleverly tricked Efetobore Sodje on the by-line, forcing the Bury defender to foul him from behind and Lee Miller hammered the subsequent spot kick low into the bottom left-hand corner.
Matty Robson extended Carlisle's lead in the 10th minute, turning adroitly near the penalty spot to drive Frank Simek's cross home left-footed after the full-back had made a raiding run down Bury's exposed right flank.
Ironically Shakers could also have scored twice in a blistering opening spell as Tom Hopper's header was pushed on to the crossbar by Carlisle goalkeeper Mark Gillespie then the on-loan Leicester forward should have done better with a David Worrall cross, which he turned wide from a good position in the box.
Bury came out much-improved after the break and it was no surprise when former Carlisle midfielder Schumacher pulled a goal back in the 53rd minute, planting a firm header past Gillespie after substitute Sebastien Carole had provided the cross from the left.
Gillespie made fine saves from Schumacher, Worrall and Carole as Carlisle clung on to their narrow advantage and again the goalkeeper came to the rescue to deny Hopper in injury time.