Carlisle picked up their first points of the season thanks to goals from James Berrett and Liam Noble.
The Cumbrians quickly banished memories of a disastrous 3-0 defeat to Notts County on the opening day by making all the early running.
Tom Taiwo hit the inside of the post after a thrilling run and through-ball from Matty Robson and Berrett rifled a long-range effort narrowly wide.
Bury took a while to get going, Ryan Lowe drawing a superb save from Adam Collin with their first chance of note.
Collin was called on again to deny Worrall, but the visitors went ahead when Berrett's speculative 20-yard shot squirmed past Bury goalkeeper Cameron Belford.
Jones had a goal chalked off for a foul in the build-up, while Lenell John-Lewis nudged a tame effort wide of the post before Noble slotted home Andrew Welsh's cut-back in the final minute to make sure of victory.
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