Burnley were knocked out of the Capital One Cup as a 78th minute penalty from Adthe Nuhiu put paid to the Clarets hopes of progression.
Substitute Nuhiu converted from the spot after Jérémy Hélan was felled in the box by skipper Jason Shackell.
The Clarets were unable to break through the visitors’ defence, and were denied three penalty appeals, meaning Wednesday go into the hat for the third round.
Sean Dyche handed competitive debuts to summer signings Matt Gilks and Stephen Ward as he made seven changes following Saturday’s narrow defeat at Swansea City.
Marvin Sordell made his first Burnley start while Kevin Long, Ross Wallace, fit-again Michael Kightly and Ashley Barnes were given their outings from the first minute, too. Scott Arfield partnered David Jones in central midfield.
Former Clarets first-team coach and caretaker manager Stuart Gray also opted for several changes to his side, with Wednesday looking to build on their unbeaten start to the season in all competitions.
It took Burnley less than three minutes to carve out their first opening.
Good combination play down the right saw Wallace cross the ball into the box, but Kightly, perhaps surprised at the amount of space he found himself in, couldn’t provide the finish at the far post.
The visitors responded moments later when first Kamil Zayatte and then Gary Madine had efforts off target following a succession of corners.
All in all though, the match got off to a rather slow start.
On 18 minutes, there was a big sigh of relief inside Turf Moor when Long diverted Liam Palmer’s dangerous ball into the box into the top of his own crossbar.
Not long afterwards, Wallace struck a free-kick over the crossbar before Barnes headed the Scot’s in-swinging cross wide of the post.
Jérémy Hélan then threatened for Wednesday, but his shot was eventually deflected comfortably into the hands of Gilks by the Clarets defence.
And the first half ended goalless as Burnley were unable to get into their rhythm.
They were clearly into their stride as soon as the game resumed after the interval, though.
First, Barnes combined with Kightly on the edge of the box with the former testing Chris Kirkland in the Owls goal.
Moments later, Sordell took centre stage as he turned neatly inside the penalty area and sent a dangerous shot over the crossbar before his deflected strike from distance was gathered not without problems by the Wednesday goalkeeper.
It was then the visitors’ turn to respond and another header by Madine saw Gilks called into action. The debutant dealt with the effort comfortably.
Matt Taylor replaced Jones on the hour-mark, but the Owls soon created another chance when Chris Maguire smashed the ball into the side-netting from a good position inside the box.
On 68 minutes, a quality cross by Ward was headed off target by Barnes at the far post as Burnley kept looking for a breakthrough.
Long also went close with the match approaching its final quarter of an hour, but he couldn’t turn the ball goalwards on a left-foot volley after Barnes had knocked the ball down on the edge of the six-yard box.
And it was Wednesday who managed to score the all-important goal.
Shackell brought down the pacey Hélan just inside the penalty area and Nuhiu calmly converted the resulting spot kick.
Danny Ings and Lukas Jutkiewicz were then introduced into action with just eight minutes remaining on the clock.
But despite their best efforts and late pressure the Clarets couldn’t force an equaliser as they suffered their third defeat in as many matches this season.
Burnley: Gilks, Trippier, Long, Shackell, Ward, Wallace, Arfield, Jones (Taylor 60), Kightly (Ings 82), Sordell (Jutkiewicz 82), Barnes.
Subs not used: Heaton, Duff, Mee, Reid.
Sheffield Wednesday: Kirkland – Palmer, Lees, Zayatte, Mattock – Semedo – Maguire (Nuhiu 73), Coke, Hélan, Maghoma (May 62) – Madine.
Subs not used: Westwood, Loovens, Floro, McCabe, Corry.
Goals: Nuhiu 78 (pen)
Bookings: Maguire, Coke
Referee: G. Eltringham
Attendance: 4,979 (1,052)