A lively contest looked to be heading for a draw until Marcus Tudgay headed his second goal of the game from Graham Kavanagh's 85th-minute corner.
And Wednesday wrapped it up two minutes later when Deon Burton's 20-yard shot took two deflections to leave Steve Simonsen stranded.
The first half was a defensive horror show and contained four goals which must have made both managers squirm.
Stoke took an 12th-minute lead when Richard Wood failed to deal with Andy Wilkinson's punt forward and Ricardo Fuller bullied him out of the way, side-stepped Michael Johnson and placed a 14-yard shot past Lee Grant.
But Wednesday were soon on level terms as the elusive Jermaine Johnson turned inside Wilkinson with ease and then had acres of space to fire in an unstoppable left-foot shot.
Stoke's rearguard was looking unusually ragged and they were carved open again when the visitors took a 23rd-minute lead.
Tudgay's initial shot struck a post, but the striker made no mistake with a six-yard header after Burton O'Brien looped the loose ball back into the danger area.
Wednesday then held the upper hand, with O'Brien almost adding a third before Lee Bullen gifted Stoke a 41st-minute equaliser.
The full-back attempted to head the ball back to Grant, but Fuller was onto it in a flash to nod past the stranded keeper.
The defensive frailties continued at the start of the second half as Tudgay blazed wide when well-placed and Fuller threatened again at the other end.
Man-of-the-match Fuller then laid on two excellent chances which were fluffed by Richard Cresswell and John Eustace during Stoke's best passage of play.
And Wednesday made them pay to lift the pressure on manager Brian Laws.