Midfielder Honeyman dinked over Shakers goalkeeper Joe Murphy with 21 minutes to go after being found by Didier Ndong to earn manager Simon Grayson his maiden victory in the Black Cats' post at their League One hosts.
Grayson included midfielder Gibson for the first time since the Black Cats instigated a disciplinary process against him after footage emerged of him criticising team-mates in a bar.
The visitors had two excellent chances early on with Lewis Grabban nodding a Brendan Galloway cross just wide and Murphy saving smartly from Aiden McGeady.
The Shakers were then forced into a 16th-minute change when skipper Stephen Dawson was carried off after appearing to injure his knee in a seemingly innocuous challenge with Gibson.
Wahbi Khazri, one of the players Gibson had been quizzed about by fans during his filmed drunken rant, looked destined to give Sunderland the lead in the second half when he rounded Murphy only to lose his footing and fail to get a shot away.
Honeyman was far more sure-footed when his opportunity came in the 69th minute as he collected Ndong's pass to the right of the goal and lifted his effort over Murphy and into the net.
Substitute Jermaine Beckford came closest to an equaliser but fired wide while Ndong could have scored himself at the other end when he rifled a half volley over.