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Nuhiu scored after just four minutes when he converted from close range to punish Boro's inability to clear their lines from a corner.
Leadbitter went close for the home side on 15 minutes when his shot clipped the top of the crossbar, before Tomas Meijas was forced to tip Stevie May's effort onto the post.
With around five minutes of the first half remaining, May struck his maiden goal for Wednesday when he flicked the ball past Meijas to double the visitors' advantage.
However, Boro did grab one back when Leadbitter converted from 12 yards on 71 minutes, and they were awarded another spot kick in stoppage time, and the midfielder made no mistake once more to set up a grandstand finish.
However the Owls held on to secure the win and move onto eight points from four games.