Rovers slipped to a narrow 2-1 loss on a night of high drama at Hillsborough.
Vincent Sasso struck a first half brace for the Owls, either side of a Sam Hutchinson own goal after he had diverted Danny Graham's header into his own net.
But in a second half that Rovers dominated they saw Ashley Westwood make a fine double save to keep out substitutes Marvin Emnes and Liam Feeney, before Emnes hit a shot that crashed in off the crossbar but the goal wasn't given.
Rovers had the ball in the net once more soon afterwards as Hope Akpan lashed home, but the referee blew for an infringement and in his frustration the midfielder pushed the official and was duly red-carded.
The Rovers pressed until the end, but it was the first half lapses from set-pieces that would prove costly.
Rovers made three changes from the side that drew with Rotherham United at the weekend as Connor Mahoney, Danny Guthrie and Hope Akpan all came into the eleven. And it was young Mahoney who caught the eye early on at Hillsborough.
The 20-year-old was involved from the off with two shots at goal inside the opening eight minutes as Rovers started strongly, but neither tested Ashley Westwood in the Owls goal.
The home side eventually posed a threat on 16 minutes when Ross Wallace rattled the crossbar from a free-kick. Two minutes later his Scottish counterpart Barry Bannan would deliver another set-piece ball into the Rovers area to where defender Sasso arrived at the back post to spectacularly volley into the back of the net.
But Rovers didn't let their heads drop, getting back on level terms almost immediately.
A poor back-pass from Morgan Fox had Westwood scrambling backwards towards his own goal, needing to tip the ball over the bar. Elliott Bennett delivered the resulting corner from the left, Graham nodded on and the ball deflected off Hutchinson and into the back of the net.
Rovers continued to enjoy the better of the play, giving their opponents very little time in possession, Mahoney had another shooting chance, but his near-post shot was saved by the Wednesday keeper.
However on 33 minutes Rovers were forced into an early chance when Gallagher was on the end of a poor challenge from Hutchinson as the midfielder caught the Rover high and late.
Gallagher was stretchered off to be replaced by Craig Conway, Hutchinson perhaps fortunate to escape with just a yellow card.
But as well as Rovers were playing in the final third, it was their inability to defend set-pieces that would prove costly and ensure they went in at the break trailing once more.
On 44 Wallace hit a corner from the right, Sasso, unmarked for the second time on the night, headed powerfully into the corner of the net.
The Owls brought on Fernando Forestieri at the start of a second half that Rovers were bright from the off once more as they forced a series of early corners. One cross to the back post from Conway was almost met by Akpan, but the ball whistled just past his outstretched boot.
Wednesday responded with two chances in quick succession as Bannan shot wide from distance before Darragh Lenihan had to clear over his bar from a dangerous cross from Adam Reach.
But aside from long-range efforts, that would be as close as the Owls would come to threatening Jason Steele's goal as Rovers pressed for an equaliser for a second time.
Conway was just wide with an effort across goal after stealing the ball from Hutchinson, before substitute Marvin Emnes fired over when well placed from a Liam Feeney pass. His blushes were saved by the linesman's flag.
With eight minutes to go Wednesday were grateful for a fine double save from Westwood. The keeper first kept out an effort from Emnes, before denying Feeney on the volley.
But Rovers kept on pressing and should have been celebrating an 85th minute equaliser when Emnes let fly from distance. The ball crashed down off the crossbar and bounced behind the line, but the officials didn't give the goal.
Aghast by that decision it then got more annoying for the visitors as they saw another goal chalked off. This time Akpan having been adjudged to have handled before blasting the ball into the back of the net.
But in his frustration at the goal being disallowed the midfielder laid his hands on the referee and was issued with a straight red card.
Against an increasingly nervy home side Rovers pushed until the end, Steele caused chaos when coming up for a late free-kick before Darragh Lenihan blasted a long-range strike just past the post.
Rovers didn't deserve to come away from Hillsborough with nothing from the match, but as well as they played, two lapses from set-pieces in the first half ultimately lead to their downfall.
SHEFF WED: Westwood, Hunt, Loovens (c), Sasso, Fox (sub Forestieri 46 mins), Wallace, Bannan, Hutchinson (sub Jones 73 mins), Reach, Winnall (sub Fletcher 56 mins), Rhodes.
Subs not used: Wildsmith (GK), Abdi, McManaman, Palmer.
Goals: Sasso 18, 44 mins.
Booked: Hutchinson 33 mins, Bannan 45+3 mins, Jones 89 mins.
ROVERS: Steele, Lowe (c), Lenihan, Mulgrew, Williams, Bennett, Akpan, Guthrie (sub Emnes 65 mins), Mahoney (sub Feeney 72 mins), Gallagher (sub Conway 37 mins), Graham.
Subs not used: Raya (GK), Nyambe, Tomlinson, Brown.
Goals: Hutchinson (OG) 20 mins.
Booked: Bennett 90+2 mins.
Off: Akpan 86 mins.
Attendance: 24,603 (including 373 Rovers fans)