It plunged them back into relegation trouble as fellow strugglers Scunthorpe climbed above them and are now six points clear of the drop zone.
Wednesday were rocked after just two minutes in this local derby when the home side snatched the lead through leading marksman Gary Hooper.
He burst through a huge gap in the Wednesday defence to steer his shot past goalkeeper Lee Grant for his 10th goal of the season.
United almost increased their lead after 34 minutes when midfield man Grant McCann rifled in a left-foot shot that was tipped over the top by Grant.
Wednesday saw plenty of possession as they searched for an equaliser but lacked a cutting edge up front.
The Owls dominated the start of the second half and had Scunthorpe on the back foot for long spells.
But when a golden chance for an equaliser finally arrived for veteran Michael Gray, his finish was a poor one as he scuffed his shot wide.
It was Scunthorpe who made the game safe by snatching a second goal after 77 minutes when McCann found the top corner at the second attempt after his first shot had been blocked.
Wednesday brought on strikers Luke Varney and Akpo Sodje from the bench in the closing stages but failed to carve out a chance to get themselves back into the game.