Loanee Harry Chapman scored his first ever Blades goal and it proved to be the winner.
United hosted Bristol Rovers for the first time for 18 years and substitute Chapman decided the tie with a second half header.
In a frantic end to the fixture, United defended stoutly to make it six games unbeaten.
Scotsman Scougall returned to the starting XI, replacing the suspended Basham, as Wilson and Wright failed to shake off injury problems. Teenager Brooks was named amongst the substitutes. The Blades reverted to 4-4-2 and kicked towards the Kop in the first half.
The visitors quickly forced themselves onto the front foot as United struggled to keep hold of the ball in the early stages. Ollie Clarke curled wide, whilst the Blades had to make two blocks following a melee from a corner.
United's first chance arrived on 16 minutes. After robbing possession in midfield, Fleck shot wide and moments later passes from Duffy and Scougall put Sharp down the left channel but goalkeeper Roos saved with his feet.
Midway through the half neither side could lay claim to getting on top but Rovers went close on 38 minutes. From a free-kick Moore made a full-length diving save to his left to keep out a Lines shot.
Like the first half, the first chance of the second period fell to Ollie Clarke, who shot wide for the visitors. Rovers went closer still on 52 minutes when Harrison's ball beat the offside trap but Easter poked wide with Moore rushing off his line.
Chapman was introduced as Wilder opted for more pace and the substitute went close just after the hour, poking wide a loose ball wide after Sharp collided with Roos from a quick Coutts free-kick.
For the visitors, Harrison shot at Moore but Chapman's next contribution was to open his Blades account. Lafferty found space on the left and from his cross CHAPMAN rose above his marker to head past Roos from six yards. He celebrated with a flip!
There was more drama from the restart, Easter jumped into O'Connell and his challenge was deemed to be a red card offence by referee Sarginson.
Moore did well to hold a shot from Lines and Rovers used their third attacking change with 15 minutes to go.
Done half-volleyed wide after cutting in from the right but the visitors were seeing more of the ball and boss Wilder opted to throw on more fresh legs with Lavery taking over from Sharp.
Late on United's intention was to keep possession to frustrate Rovers and for the most part the home side were successful. It was the five minutes of injury-time that brought the drama.
First of all, Fleck cleared off the line from Harrison following a corner and from a break, Leadbitter produced a cynical foul on Lavery as Roos had added himself to the attack for the corner.
Brown was introduced for his debut and late on Freeman and Ebanks-Landell got in each other's way but Lafferty cleared before the clean sheet and victory was confirmed.
Blades: Moore, Fleck, O'Connell, Scougall (Chapman 54), Sharp (Lavery 84), Done, Coutts, Freeman, Ebanks-Landell, Duffy (Brown 90), Lafferty.
Unused: Hussey, Ramsdale, Whiteman, Brooks.
Rovers: Roos, Clarke-Slater, Lockyer, J. Clarke (Bodin 62), Brown (James 74), Boateng (Moore 62), O. Clarke, Leadbitter, Lines, Harrison, Easter.
Unused: Mildenhall, Colkett, Montano, Gaffney.