There were few moments of quality in a low-key affair, and the visitors were able to continue their strong recent record at Burnley as the big striker made the hosts pay after skipper Jason Shackell had brought down Jeremy Helan inside the area.
The win means Wednesday are now nine matches unbeaten at Turf Moor, while for Burnley it was another setback after defeats in their two opening games back in the Premier League.
Wednesday, by contrast, are yet to taste defeat during 2014-15, with former Burnley caretaker boss Stuart Gray enjoying his return to his old club.
The visitors had come closest to opening the scoring in the first half when Kevin Long deflected the ball onto his own crossbar, and Burnley struggled to create any chances of note in the second period as Sean Dyche's men were sent crashing out of the competition at the first hurdle.
Dyche made seven changes from the 1-0 defeat at Swansea City on Saturday, Stephen Ward making his debut at left-back while the first-choice strike duo of Danny Ings and Lukas Jutkiewicz were named on the bench.
Wednesday did likewise with their front pairing as Gary Madine and Helan came in for Nuhiu and Stevie May as part of four alterations from the weekend’s 3-2 win at Middlesbrough.
Michael Kightly fluffed an early chance for the hosts by missing his kick in front of goal, before Madine headed Chris Maguire's corner over the bar in the seventh minute.
Burnley were fortunate not to fall behind in the 18th minute when Long diverted Maguire's teasing delivery onto the bar and behind for a corner.
Ashley Barnes threatened the visitors' goal six minutes later, but the forward's header from Ross Wallace's cross flew narrowly wide.
The Premier League side began the second period brightly with Barnes and Marvin Sordell both going close inside the opening five minutes.
Wednesday continued to pose a threat and went close to breaking the deadlock when Maguire fired into the side-netting on the hour-mark.
Barnes continued to make his presence felt and headed Ward's left-wing cross off target midway through the half.
The key moment arrived in the 78th minute as Shackell brought down Helan, and Nuhiu calmly sent the spot-kick down the middle.
