The Owls slipped to a 3-1 defeat at Middlesbrough and have two games to go against Nottingham Forest and Derby to beat the drop.
Yannick Bolasie fired Boro into the lead before Josh Windass equalised from the penalty spot in an incident-packed first half.
But despite bossing long periods of the game, the Owls conceded twice after the break as substitute Josh Coburn and Duncan Watmore made the points safe.
Wednesday made the early running at the Riverside but it was Boro that fashioned the first real opening when Patrick McNair drilled across Keiren Westwood and wide.
Barry Bannan shot into the midriff of Jordan Archer moments later before the hosts burst into the lead 20 minutes in.
Julian Börner had his pocket picked by Bolasie, who motored forward on the angle to finish high into the net with the aid of a deflection.
Boro had their tails up and George Saville headed a set piece over but Wednesday regrouped and threatened themselves, with Andre Green going agonisingly close with a snapshot from just inside the area.
Westwood spectacularly denied McNair with a flying save and Bolasie almost doubled his tally after cutting in to flash a low shot past the post.
But with the home side starting to turn the screw, the Owls broke and won a penalty from which they levelled.
Green embarked on a tricky run into the box that was only ended by Marc Bola, who upended the former Villa man.
The referee pointed to the spot and Windass gave Archer no chance from 12 yards.
Wednesday, though, suffered an immediate blow when Tom Lees landed awkwardly from an aerial challenge and lied prostrate on the turf before being stretchered off after a lengthy delay.
The first half ended with Archer diving to keep out an Adam Reach hit before Liam Palmer hooked over the bar right on the whistle.
Wednesday continued on the front foot and Windass sidefooted wide following excellent work down the flank by Green.
Dominant and confident in possession, the Owls looked by far the more likely to make the next breakthrough.
But against the run of play, the home side restored their lead on 74 minutes.
Dangerman Bolasie got to the byline and delivered a hanging cross that Coburn glanced past Westwood off the far post.
Ex-Boro striker Jordan Rhodes headed into the grateful arms of Archer from a Green cross straight after the restart as the Owls sought an instant way back.
But those hopes were dashed eight minutes from time when Watmore stabbed home from close range to put the contest to bed.
Wednesday: Westwood, Palmer, Hutchinson, Lees (Urhoghide 41), Börner (Hunt 82), Bannan, Pelupessy, Reach, Green (Harris 82), Windass, Paterson (Rhodes 71)