An early injury to midfielder Stephen Dawson proved the main talking point for a largely uneventful first half.
Honeyman found the all-important goal on 69 minutes with a meticulous lob over Joe Murphy to send the away side through to the second round of the Carabao Cup.
Lee Clark opted for three changes to the side that collected all three points on the first game of the season against Walsall. Debutants Alex Whitmore and Harry Bunn started, while Tsun Dai made his full professional debut in front of the Sky cameras.
In an attempt to nullify the potential threat from the away side, Clark converted to a 5-2-1-2 formation, with the inclusion of Whitmore emphasising the importance of consecutive clean sheets.
The visiting side wasted no time in the opening stages when Brendan Galloway delivered an early cross to the front post for Lewis Grabban, but his headed effort fell short of the target.
With four minutes played, Sunderland’s summer signing Aiden McGeady was denied by the fast reactions of Joe Murphy, as the Shakers goalkeeper dropped to his left-hand side to prevent an early lead for the Championship side.
Andrew Tutte came off the bench with just 16 minutes played, after Dawson was stretchered off with a nasty leg injury following a robust challenge on Darron Gibson.
An audacious first-time effort by Wahbi Khazri from 25 yards out came close to breaking the deadlock, but his effort soared uncomfortably over the crossbar.
Patient passing play by the Shakers forced Simon Grayson’s side onto the back foot, as new signing Bunn switched the play to an unmarked Craig Jones down the right, with the latter struggling to hit the target.
Last season’s Player of the Season for the Shakers, James Vaughan, returned to Bury, this time for the Black Cats, as a half time substitute, and a straight swap for striker Lewis Grabban.
Khazri had an immediate opportunity after the break to send the visiting supporters behind the goal into raptures, but he wasted his chance despite taking the ball around Murphy, he slipped and watched on as the ball rolled out for a goal kick.
Zeli Ismail’s introduction for Adam Thompson allowed an alteration to the formation, converting to 4-4-2, and a sign of attacking intent from Clark going into the last 30 minutes.
The travelling supporters were eventually celebrating thanks to a 69th minute goal from George Honeyman, his first for the club, as he chipped the ball over Murphy and into the back of the net from inside the penalty box.
Substitute Jermaine Beckford flashed an effort toward goal from 30 yards-out in an attempt to catch Jason Steele off guard, but he was unsuccessful in his desperate effort with twelve minutes remaining.
Beckford again demonstrated his attacking prowess with a deft touch from a Craig Jones cross, although he was unable to divert it goalward at the front post.
The Shakers return to league action this Sunday as they prepare to face newly relegated Wigan Athletic at the DW Stadium.
Attendance: 3,470 (934 travelling supporters)
Bury starting line-up: Murphy, Jones, Thompson (Ismail 59), Aldred, Whitemore, Leigh, Reilly, Dawson (Tutte 16), Dai (Beckford 72), Bunn, Ajose
Unused subs: Humphrey, Cameron, Fasan, Lowe
Sunderland starting line-up: Steele, Matthews, Browning, O’Shea, Galloway, Gibson, N’Dong, Honeyman, McGeady (Asoro 59), Khazri (Gooch 76), Grabban (Vaughan 45)
Unused subs: Rodwell, Love, Kone, Ruiter