Second-half goals from James Berrett and Liam Noble saw Carlisle United pick up their first points of the season.
The Cumbrians were the better side early on, Tom Taiwo hitting the post after a fine run and pass from Matty Robson and Berrett going close.
Bury responded with Ryan Lowe forcing a superb save from keeper Adam Collin.
But Berrett's speculative 20-yard shot beat keeper Cameron Belford to put the visitors ahead and Noble slotted home in injury-time to seal the victory.
VIEWS FROM THE DRESSING ROOM
Bury manager Richie Barker told BBC Radio Manchester:
"There are very, very few positives to take from it, and I was disappointed with the result but more disappointed with the performance.
"If I'm honest they just got round the pitch and worked harder than we did.
"I can accept the fact we can get beaten by better teams, but we just didn't match them today."