First half goals from Joe Bunney and a brace from Brad Inman gave the visitors a three-goal advantage at the break.
Ian Henderson scored the only goal in the second half as the Shakers started the competition with a defeat.
Lee Clark made eleven changes to the team that drew with Plymouth Argyle at the weekend, with Ryan Lowe leading from the front as captain.
Bunney opened the scoring after just 12 minutes with a darting run into the penalty area, before sliding the ball into the bottom left hand corner of the goal.
The visitors did not take long to find the back of the net again, this time through Inman, who finished in the same fashion to the first goal just five minutes earlier.
Two minutes later, Bury’s Jordan Williams went in search of an immediate comeback. His run down the right carved open the defence, his pull back found Lowe, but the Liverpudlian’s first-time shot rebounded back off the right-hand post.
The Shakers continued to threaten the visitors down the right flank, the skilful Mihai Dobre blasted the ball goalwards, his shot was saved and Neil Danns’ aimless first time shot flew over the crossbar.
On 34 minutes, Inman found himself on the end of a direct ball over the defence and finished the ball to the left of Leo Fasan to make it 3-0.
With a quiet end to the first half, finishing at 3-0, the Shakers were forced into three changes at half time: Greg Leigh, Callum Reilly and Harry Bunn replacing Jordan Williams, Alex Whitmore and Callum Styles.
Six minutes after the break and the away side found their fourth goal of the game. Henderson’s forward run down the left wing afforded the forward time in front of goal, before finishing past Fasan as the ball bounced off the post and into the back of the net.
Dobre came close to pulling a goal back for Clark’s side with 62 minutes played. His deadly in-swinging free kick from the left flank was denied by the fingertips of Josh Lillis in the Rochdale goal.
The remainder of the second half gave little for either supporters to cheer about as the visitors grabbed all three points to take them to the top of Group C.
Attendance: 1,451 (498 travelling supporters)
Bury: Fasan, Williams (Bunn 45), Whitmore (Leigh 45), Shotton, Skarz, Styles (Reilly 45), Dai, Cooney, Danns, Dobre, Lowe
Unused subs: Maloney, Hill, Harker, Hulme
Rochdale: Lillis, Rafferty, Ntihe, McGahey, Bunney, Camps, J. Williams (Rathbone 73), Inman (Cannon 69), Adshead, J. Williams, Henderson (Slew 55)
Unused subs: Moore, Morley, Gillam, Kearney