Fox opened the scoring just before half-time after a tepid opening to the match, with the left-back found in acres of space from a quick Sam Hutchinson throw-in.
QPR improved in the second half and could have had an equaliser on a couple of occasions but Winnall put the tie beyond doubt in the 90th minute with a calm finish from six yards out.
Nakhi Wells pulled one back in injury time following a failed Cameron Dawson clearance, but it was too late for the home side to find a second.
Owls dominate early proceedings
The Owls started slightly the better of the two sides in possession, but failed to create any clear cut chances with their early forays into the QPR half.
The home side slowly grew into the contest and created the first real chance of the game on 15 minutes through midfielder Marc Pugh, but his right-footed shot from the edge of the Wednesday area was blocked.
Minutes later it was Wednesday’s turn to go close releasing Jacob Murphy down the right hand side, but the former Norwich winger’s shot was saved by QPR goalkeeper Joe Lumley at the near post.
As the half wore on play became increasingly scrappy, with both sides committing cynical fouls in order to prevent attacks from progressing.
However, with just two minutes to go until the break, Fox gave Wednesday the lead after a quickly taken Hutchinson throw-in.
Hutchinson’s ball found Fox wide open in the QPR area and the full-back’s left-footed shot squirmed through the grasp of Lumley to put the away side ahead.
Mark Warburton would have been furious with his side for switching off so close to the break, with no one tracking Fox’s run into the box.
QPR improve after the break
With no changes at half time it was Garry Monk’s side who continued to create the majority of the openings early on, with Rangers twice repelling dangerous crosses from Murphy within five minutes of the restart.
In a blow for the away side Hutchinson was forced to hobble off inured after 53 minutes, with Alex Hunt replacing the former Chelsea man in midfield.
The change made little impact on Wednesday though, with the away side comfortably keeping QPR at arm’s length.
In response, Warburton replaced on-loan wide man Jack Clarke with Bright Osayi-Samuel and the change nearly had an instant impact with Eberechi Eze forcing Dawson into a save from the edge of the area with the home side’s first attempt on target of the match.
Minutes later QPR went close again when Osayi-Samuel set up Todd Kane, but the full-back shot wide from 12 yards out.
With 20 minutes to play, QPR wasted another opportunity when Elias Chair shot wide from the edge of the box under very little pressure from the Wednesday defence.
The R’s were very much on top by this stage and Jordan Hugill missed a guilt-edged chance from six yard out after a fantastic ball from Kane.
With the home side chasing an equaliser, Warburton sent for top scorer Wells, who replaced Chair, while Monk brought on Adam Reach for Murphy in an attempt to shore up his midfield.
QPR’s dominance continued though and Eze was the next to go close when his shot was blocked following excellent work from Osayi-Samuel on the right wing.
With the home side pushed forward Atdhe Nuhiu nearly caught out Lumley on the counter attack after the goalkeeper strayed too far away from his line, but the stopper was just able to recover the potentially embarrassing situation.
Lumley’s resulting clearance sprung a counter attack and Eze once again found himself open in the area, but this time his shot was straight at Dawson in goal.
QPR had a late penalty shout turned down when Pugh went down inside the area, but Wednesday sealed the game on the counter with a 90th minute Winnall goal.
Reach was set free down the left hand side after a Nuhiu cross field pass, with the former Middlesbrough winger squaring for Winnall to put the tie beyond doubt.
Wells did grab a late consolation for Rangers after Dawson misplaced a clearance, with the Bermudan striker rounding the goalkeeper to make it 2-1.
Rangers then pushed up for the final minute and won a later corner, but the home side were able to force an equaliser and send the match to a replay.