City, 17th in the table but unbeaten in their last three league matches, dominated the first half and appeared set for a comfortable passage, but goals from Wednesday trio Gary Teale, Neil Mellor and Clinton Morrison after the break turned the game on its head.
The Owls' promotion push in League One had suffered a setback following successive defeats, but once winger Teale gave the visitors a shock lead early in the second half, Keith Millens' Robins fell apart.
Owls striker Mellor headed a second in the 55th minute and when Morrison despatched the third goal nine minutes later, the game was over as a contest.
City, unchanged from the side that started in Monday's 2-2 draw at QPR, were dealt an early blow when defender Steven Caulker, on loan from Tottenham, was forced off through injury and replaced by Liam Fontaine.
But the Robins dominated the first period, with Albert Adomah and Brett Pitman shooting wide before Marvin Elliott's header from Jamal Campbell-Ryce's corner was just off target.
Wednesday's travelling fans had to wait until the 19th minute before their side's first effort on goal, Morrison pulling his 25-yard shot wide.
Owls manager Alan Irvine included new signings Reda Johnson and Michael Morrison in a new-look back four - Daniel Jones was also recalled - but Elliott caused problems again from another Campbell-Ryce corner, this time heading inches over the crossbar.
Pittman, Jonathan Stead and Elliott all threatened to break the deadlock in City's favour before half-time.
But Wednesday snatched a shock lead four minutes after the restart when Chris Sedgwick crossed from the right for Teale to head home from six yards.
City were stunned, while the visitors visibly grew in stature after such a turnaround.
Wednesday pressed forward again and went two up six minutes later when Jones crossed from the left and Mellor pounced to head home into the bottom corner.
Sedgwick blazed over the crossbar from close range soon after and with City looking increasingly jittery every time Wednesday countered, Morrison added a game-clinching third goal.
Mellor was this time the provider, sweeping the ball across the six-yard box at the end of a flowing move in the 65th minute for Morrison to sidefoot beyond City goalkeeper David James.
Fontaine headed over the crossbar for City and both Stead and Pitman forced fumbling saves from Owls goalkeeper Nicky Weaver, but it was too little too late for Millen's side.