George Boyd looked to have earned the Owls all three points, with his 78th minute goal, but up popped Wilbraham late in the game to head home.
Sean Clare was brought back into the Wednesday starting XI for the crunch Championship game with Bolton.
The midfielder missed the midweek defeat to Ipswich with an ankle injury, but returned to the Owls midfield alongside Joey Pelupessy and Adam Reach.
Lucas Joao started up front alongside Atdhe Nuhiu, with Jordan Rhodes dropping to the bench.
There was also a return in the back five for Jack Hunt and Daniel Pudil.
Nuhiu headed wide from an early chance, but it was a low key start for both teams.
Owls defender Tom Lees was forced into a saving tackle to deny Adam Le Fondre, after a break from Sammy Ameobi, while at the other end Pelupessy saw his effort blocked.
Nuhiu has Wednesday's best chance, after good work from Reach, but after beating two defenders, goalkeeper Ben Alnwock produced a fine save to keep the game scoreless.
At the other end, Joe Wildsmith produced a save of similar calibre, diving low to his right to keep out Will Buckley's far-post header.
Chances were at a premium in the second half. Joao's cross just evaded Reach, while Nuhiu's header drifted wide.
But the Owls finally grabbed a goal on 78 minutes.
George Boyd drilled the ball beyond Alnwick, and despite a linesman's flag - presumably with Jordan Rhodes in an offside position - the referee awarded the goal.
But in four minutes of stoppage time, the Owls conceded a needless free-kick, and when the ball was whipped in, substitute Aaron Wilbraham headed in an equaliser.