The Kosovo international found the net before and after the interval to put the Owls on course for round five, before George Boyd scored on the hour to ensure that Cameron Dawson's late own goal counted for little.
Reading had the chance to open the scoring in the fourth minute when David Edwards gained possession inside the box and let fly, forcing an early save out of Wednesday goalkeeper Dawson, but it was otherwise a subdued first half until the hosts took the lead with half an hour played.
Morgan Fox teed up Nuhiu inside the opposition's penalty area and the striker kept his cool, rifling the ball into the bottom corner to send to Owls into the break with a slender lead.
Jaap Stam's men had two chances to draw level early in the second period, with Dawson saving Wednesday's skin on both occasions, pulling off stops to deny Yann Kermorgant and, moments later, Jon Dadi Bodvarsson.
The home side did not have their backs to the wall for long as Nuhiu turned in his second from close range in the 53rd minute, guiding Marco Matias's low pass beyond Anssi Jaakkola to give his side breathing space.
Less than 10 minutes later, Wednesday put the result beyond doubt when Adam Reach's cross into the penalty area found the Reading defence ball watching, allowing Boyd to slot home with Jaakkola out of position.
The Royals netted a consolation five minutes from time when Frederico Venancio's strike cannoned off the post and ended up in the net via a combination of Bodvarsson and Dawson, with the Wednesday keeper providing the touch which guided the ball over the line.
The last time Wednesday defeated Reading in the FA Cup was in 1966, and that year they went on to reach the final of the competition.
WEDNESDAY (5-3-2): Dawson, Loovens, Pudil (Palmer '63), Venancio, Fox, Hunt, Pelupessy, Reach (Jones '71), Boyd, Nuhiu, Matias (Rhodes '89)
READING (4-4-2): Jaakkola, Tiago, Moore, Bacuna (Evans '79), Gunter, Edwards, Clement, McCleary (Barrow '66), Beerens, Bodvarsson, Kermorgant (Smith '79)