Substitute Nuhiu slotted home his fourth goal of the season from the spot in the 87th minute after Dean Whitehead was sent off for handling Leon Best's goalbound shot.
Earlier, Middlesbrough had seen their first goal in seven matches ruled out when Jacob Butterfield's strike hit both posts before appearing to cross the line.
Boro forward Kei Kamara had the first opportunity of the game but his fourth-minute volley from Albert Adomah's right-wing cross was comfortably saved by Owls goalkeeper Chris Kirkland.
The visitors then thought they had taken the lead when Butterfield's drive cannoned off both uprights and looked to have crossed the goal line before the ball was grabbed by Kirkland - but no goal was given.
Wednesday should have gone ahead on the half hour mark through on-loan striker Best. Full-back Liam Palmer played a one-two with Chris Maguire on the right before crossing to the near post where the unmarked Best, seeking the second goal of his loan spell from Blackburn, poked the ball the wrong side of the right-hand upright.
The Owls felt they should have been awarded a penalty a minute before the interval when Boro captain Jonathan Woodgate appeared to clatter midfielder Sam Hutchinson but referee Andy D'Urso dismissed the home side's appeals.
Middlesbrough's debutant keeper Tomas Mejias then had to be alert to turn Hutchinson's header behind for a corner following Maguire's free-kick into the box in first-half stoppage time.
The hosts began the second half the brighter of the two teams and lively wide-man Maguire sent a free-kick over the crossbar before Spanish keeper Mejias acrobatically saved Hutchinson's powerful long-range drive.
Boro's Danny Graham had a fantastic opportunity to open the scoring on the hour mark. The striker, searching for his first goal since joining on loan from Sunderland, raced clear of the Wednesday backline but was denied by a strong hand from Kirkland before the former England international goalkeeper tipped Kamara's follow-up effort from the edge of the area over the bar.
Owls midfielder Giles Coke was denied by Mejias in the 78th minute before the game's decisive moment arrived.
Best stabbed the ball towards goal and saw his effort repelled by the arm of midfielder Whitehead, who was dismissed, and Nuhiu, a 71st minute replacement for Maguire, calmly found the bottom right-hand corner, sending Mejias the wrong way.