Raphael Nade gave Carlisle a deserved victory with his first ever Football League goal.
The Cumbrians' summer signing from non-league Woking struck after 57 minutes of a one-sided game at Gigg Lane.
Set up by the pace and directness of Brendan McGill, the Frenchman cleverly chipped Neil Edwards from the edge of the area to make United's pressure finally tell.
Edwards, as he has done on a regular basis this season, saved the Shakers in the first half.
He denied Derek Holmes and Karl Hawley with fine saves while team-mate Paul Scott cleared off the line from Hawley and McGill, as Bury's defence was repeatedly stretched.
The match should have been over by half-time but even after the interval there was no change in Bury's chronically bad display.
Only after Nade's goal did the Shakers look likely to score. They thought referee Jonathan Moss should have awarded a 67th minute penalty when Danny Livesey clearly handled a cross from Simon Whaley.
Inexplicably, Mr Moss and his assistants did not see any offence, much to the amazement and anger of the Bury side, with Scott earning a caution for his protests.
Bury pressed forward in an attempt to salvage something from the game but Paul Arnison produced a stunning clearance from another Whaley centre with Dwayne Mattis poised to slide in the equaliser.