An eventful second-half played out in a snowstorm for long spells saw Nuhiu put the Owls in front from close range with 19 minutes to go, firing in a rebound after Adam Reach's header struck the post following Jack Hunt's centre.
Leeds dug deep and levelled with a milestone first goal for the club from substitute Jay-Roy Grot, who headed home following Pablo Hernandez's cross.
But the Owls - without a win in their previous seven matches and last eight on the road - had the final say when Nuhiu evaded Matthew Pennington before coolly steering the ball low past Bailey Peacock-Farrell - his sixth goal of a productive new year.
The strike deflated the vast majority of the 31,638 crowd, with Leeds poor recent run continuing with the hosts having won just once in eight matches under the watch of Paul Heckingbottom.
One signficant change for the Owls saw Sam Hutchinson make his long-awaited return to the starting line-up for the visitors after being out with injury since December 30.
It was a half that produced sporadic flashes of animation for both derby rivals, but plenty of scrappy play and misplaced passes too in keeping with their lamentable and thoroughly underwhelming run of form this year.
Leeds - minus the injured Samuel Saiz - were heavily reliant in the midfield axis of Pablo Hernandez, Gianni Alioski and Adam Forshaw, who dovetailed neatly on occasions.
A brilliant free-kick routine involving Hernandez and Forshaw proved comfortably the highlight from a home perspective, only to be instantly spoiled by Caleb Ekuban fluffing his lines after benefitting from the moment of brilliance on 25 minutes.
The striker fired straight at Joe Wildsmith when he really should have scored and a wild half-volley over the top soon after pointed to his confidence having taken a hit.
A fine saving challenge from Frederico Venancio to deny Alioski also provided a talking point in a decent home spell either side of the half-hour mark.
Wednesday had some encouragement themselves with Adam Reach providing their main source of inspiration.
The creative midfielder went close twice, first being denied by the legs of Bailey Peacock-Farrell early on after good play by Lucas Joao before the United keeper - again afforded encouragement - turning away a raking drive.
Reach was also involved in a key moment before the break when his free-kick on the right was headed across goal by ex-Whites defender Tom Lees with Hutchinson turning the ball home from close range - only for an offside flag to be instantly raised.
A blizzard which arrived at the interval made conditions deciededly tricky at the start of the second half, but it was Leeds, who settled the quicker.
Alioski, consistently involved for Leeds, fired over before plasying his part in a decent move which almost ended in a clear sight of goal for Ekuban, only for the Owls to clear in the nick of time.
At the other end, Hutchinson fired an effort over on the break before Leeds suffered a genuine scare with Nuhiu's curler kissing the top of the bar from distance.
The big striker fared better with just over twenty minutes to go, converting the rebound after Reach's header struck the post.
Wildsmith was called into action to deny Hernandez not too long after before the snowstorm returned to make the game somewhat of a lottery.
Leeds, to their credit, kept persevering and were rewarded when Grot headed home from close range following Hernandez's cross, but the twist in the tale came from Nuhiu.
Leeds United: Peacock-Farrell; Berardi, Jansson, Pennington, Pearce (Grot 77); Forshaw, O'Kane (Phillips 78); Alioski, Hernandez, Dallas; Ekuban (Lasooga 70). Substitutes unused: Wiedwald, Anita, Vieira, Sacko.
Sheffield Wednesday: Wildsmith; Venancio, Lees, Pudil; Hunt, Hutchinson (Bannan 66), Pelupessy (Fox 90), Reach, Boyd; Nuhiu, Joao. Substitutes unused: Dawson, Palmer, Jones, Butterfield, Rhodes.
Referee: K Stroud.