First-half goals from striker Ryan Lowe, who struck his 50th goal for the club, and Damian Mozika were enough for the League One newcomers, while Chris Sedgwick provided the only response from Gary Megson's promotion favourites at Gigg Lane.
Bury surged in front within seven minutes when Lowe headed home from David Worrall's pinpoint cross.
Andy Bishop came close to extending the hosts' lead with a glancing header, while Lewis Buxton pulled the Owls' best chance harmlessly across the face of goal.
The hosts' dominance paid off four minutes before the break when Mozika picked up a throughball from Joe Skarz and slotted into the corner.
Reda Johnson had a goal chalked off for a foul on Cameron Belford, but the Owls pulled one back after the break when Chris Sedgwick curled a delightful 20-yard effort in off the crossbar.