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And with the second half just seven minutes old it was Wilcox, also scoring his first goal in City colours, who netted the decider as strikes from Chris Morgan and Jonathan Forte proved mere United consolations.
Wilcox had initially been named among the substitutes but was promoted to the starting line-up after Kevin Harper, the on-loan Portsmouth midfielder, fell victim to a calf injury during the warm-up.
But the re-arrangement did the hosts little harm as they took the lead from their first attack which, in the eighth minute, was initiated by goalkeeper Kevin Pressman who was allowed to advance outside his box before launching the ball forward.
Dion Dublin provided the crucial headed flick and Scowcroft to ran from deep inside the right channel to slot past the exposed Paddy Kenny.
United went close to restoring parity when Ashley Ward produced an intelligent ball over the top of the Foxes defence but Nick Montgomery could only fire into the side-netting.
And Leicester went further ahead in the 28th minute thanks to Dabizas, who met Jordan Stewart's corner with a powerful header.
Initially the ball found only the crossbar but the ricochet then appeared to rebound off the Greek defender and - according to referee's assistant Wayne McIntosh - over the line.
In the second half Sheffield boss Neil Warnock brought on the attack-minded Jonathan Forte for left-back Alan Wright but the Foxes' advantage was further extended in the 52nd minute when a heavy penalty box touch from Phil Jagielka was severely punished by Jason Wilcox, who pounced on the loose ball to surgically slot the hosts' fifth of the evening.
Forte lost his bearings - and his composure too - when, in the 57th minute, Montgomery found him unmarked at the back post only for the wide man to blaze over.
In the 64th minute Blades boss Warnock opted to swap Andy Gray with Jack Lester, who found himself immediately embroiled in the action, curling an effort just wide of the far post.
With little more than 15 minutes remaining Morgan ignited hopes of a comeback when he advanced from the centre of the United defence to meet Leigh Bromby's long throw from the right with a glancing header which squeezed home at the far post.
With six minutes of regulation time remaining Forte pulled back another for the visitors, this time Lester creating havoc in the penalty area before Forte lashed the loose ball through a thicket of defenders and between the legs of unsighted Pressman.