Substitute Nuhiu rifled home to settle the second-round tie at a half-empty Turf Moor after Jason Shackell had upended Jeremy Helan.
The Clarets had been in the ascendancy at the time but the contest overall was fairly even and the Owls had gone close when Burnley’s Kevin Long struck the frame of his own goal.
Burnley wanted a penalty themselves in injury time when Ashley Barnes went down but nothing was given.
The final whistle was greeted by a roar from the away fans, who had made the most noise all night with many Burnley supporters having decided to stay away.
The match, coming between home Premier League games against Chelsea and Manchester United, was always likely to be a hard sell for the home club.
While a packed Turf Moor was rocking last Monday, the top tiers of two stands were empty on this occasion and there were plenty more vacant seats in an echoey atmosphere.
That seemed to have an effect on the pace of the game, as did the seven changes made by Burnley boss Sean Dyche, whose alterations included debuts for summer signings Matt Gilks and Stephen Ward.
Opposite number Stuart Gray – who had a brief spell in caretaker charge of the Clarets almost four years ago – included former England goalkeeper Chris Kirkland among his four changes.
Michael Kightly should have given Burnley an early lead but mis-kicked in front of goal before former Hull defender Kamil Zayatte poked wide at the opposite end.
Wednesday spurned another good opportunity when Gary Madine headed narrowly over from a Chris Maguire corner.
Burnley and Long then had a major let-off when the defender slid in to cut out a Maguire cross and turned the ball against his own woodwork for what was almost a spectacular own goal.
After that the Clarets began to settle and dominate possession.
Ross Wallace, who earlier shot at Kirkland from distance, had a free-kick deflected over.
Barnes should have done better when picked out by Wallace but headed wide from in front of goal.
Scott Arfield had a speculative effort from a free-kick inside the centre circle but Kirkland saved comfortably enough.
Gilks might have been extended before the break after a Jeremy Helan run but a deflection took the pace out of his shot.
Burnley tried to up the tempo after the restart with Barnes testing Kirkland and Marvin Sordell shooting over on the turn.
Wednesday enjoyed a good spell with Madine heading at Gilks and Maguire shooting into the side-netting.
But chances started to open up for Burnley as substitute Matt Taylor shot wide after a goalmouth scramble and Barnes missed the target with a header.
It came against the run of play when Wednesday got their penalty as Shackell fouled Helan on the right of the area.
Nuhiu fired straight down the middle for his third goal in two games as Gilks dived to his right.
Burnley produced a late flurry but not enough to trouble Kirkland.