With both sides desperate for a win it was often an edgy affair although City, despite giving away a late goal, were comfortable enough.
A pass from Michael McIndoe, as sweet as any striker could wish for, brought Adebola his goal.
It arrived perfectly in Adebola's stride and gave him time to draw defenders before knocking it past them with his left foot into the bottom right corner.
It set nerves jangling in the Owls' defence but as City failed to find a follow up they gradually pulled their game together.
The revival began when Dean Burton came in from the left with an angled shot that Adriano Basso had covered at his near post, but Wednesday's best chances were both from dead balls 30 yards out.
Each time Graham Kavanagh pulled his free-kick wide of Basso's right post, the first only inches away from success.
United team work saw the home side through a difficult patch with Lee Johnson twice diving in with relieving tackles on the edge of the box.
The busy McIndoe was also helping out in defence although this cost his side something in attack where Adebola, after that early peach of a pass, was given an indifferent service.
Still City's Australian signing, Nick Carle, will know he should have scored twice in the first half notably from Ivan Sproule's right-wing cross which left him with a clear shot.
With Sean McAllister on for the second half, Wednesday had more pace and wit and only a good save denied the substitute a goal from a header.
The game virtually escaped them though when City's forthright full-back Bradley Orr came down the right to fire home his season's fourth goal.
Lee Bullen volleyed home from a free-kick for Wednesday's consolation stoppage-time goal.