Poor form at Hillsborough has haunted Wednesday all season - they have the second-worst record on their own turf in the division - while the Blades have been the best away side in the competition, this being their sixth consecutive away win.
Wednesday carved out the better chances in a tight and occasionally edgy derby, but United are grinding out results - the win means they have lost just once in their past nine league games.
Danny Rohl's side began the stronger and when Ryo Hatsuse whipped a cross onto the head of Smith, Cooper brilliantly pushed his header over the bar.
With some bad blood lingering from United's win in the Bramall Lane derby in November, the game was tetchy, with six yellow cards, and tempers flared into a pushing match between players after Callum Paterson kicked the ball at Ben Brereton Diaz as the United striker lay prone following a foul by Barry Bannan.
Tyrese Campbell was United's biggest danger, dragging one attempt across the face of goal and then being denied by keeper James Beadle's sharp dash off his line after Gustavo Hamer's slick pass threatened to free the visitors' top scorer.
Campbell was also inches away from a Burrows drilled cross-cum-shot and in the second half a Smith back=heel teed up Windass, only for Burrows to hurl himself in the way of his goal-bound shot.
Campbell's next involvement was decisive, however, as he raced onto a pass from Sydie Peck and ghosted past Michael Ihiekwe. His cross deflected kindly into the path of Brewster, who could not miss and duly chalked up his third goal of the season.
Wednesday tried to rally but their only real chance went begging when Marvin Johnson's volley was parried by Cooper straight to Svante Ingelsson. The substitute had to make instinctive contact as the ball flew at him, and he could only send it high over the bar.