The 20-year-old, standing in for the injured Graham Kavanagh, struck in the 58th minute to finally break the stubborn resistance of an Owls side struggling at the wrong end of the Championship table.
An injury-hit home side, showing six changes to the team beaten at Ipswich last week, were well on top without being able to turn their territorial advantage into goals in the first half.
Dwight Yorke set the tone with a shot on a turn that failed to trouble Brad Jones, and a centre from Nyron Nosworthy was headed over by Tobias Hysen.
Stanislav Varga headed against the post and Liam Lawrence also failed to hit the target but many of the Black Cats' best moves broke down with several offside decisions to the irritation of the home fans.
Hysen took a shrewd pass from Yorke in his stride, his shot-cum-cross going behind off the top of the bar with Jones struggling to intercept.
In a rare Wednesday attack Deon Burton broke clear only to shoot wide and then Chris Brunt was way off-target with his effort.
Sunderland increased the tempo without being able to break down a stubborn Owls rearguard although Lawrence was just off target after cutting inside. Yorke was guilty of squandering the best chance shooting well wide with the goal at his mercy.
Sunderland continued to diminate with Dean Whitehead running 30 yards before cracking a low drive against the upright with Jones beaten in the 47th minute and, three minutes later, Varga squandered a great opportunity when heading over from Hysen's free-kick.
But the Black Cats were not to be denied and finally went ahead in the 58th minute. Nosworthy send Lawrence clear on the right and, from his low centre, Leadbitter swept the ball home at the far post.
Tommy Spur headed over in the 62nd minute and three minutes later Wednesday brought on Wade Small for an out-of-sorts Burton with Liam Miller replacing Hysen for the home side.
As the visitors hit back Ben Alnwick was forced to make a fine save from Marcus Tudgay's downward header.
Suddenly the Owls, sensing all was not lost, threw caution to the wind and, but for the brilliance of Alnwick would have snatched a late point.
The England Under-21 international produced a stunning save from Burton O'Brien and Danny Collins threw himself at the loose ball to deny Madjid Bougherra.