An impressive first half goal gave Bury the victory in Tuesday night's scrappy encounter.
In the fixture re-arranged from January, the closest the Blades come was a shot from Che Adams, which was kicked off the line.
Adkins made an enforced change for the game with Basham taking over from Collins, who picked up an ankle injury at Doncaster. Leading scorer Sharp was deemed fit enough for the bench.
On a cold and wet night, Bury forced two chances and a corner inside the first minute but the shots from Tutte and Mayor were both blocked. The early pressure came from the home side but Long's only serious action of the first 20 minutes was a catch from a Jones volley.
United's first chance was from a Baxter free-kick but Edgar's header looped onto the roof of the net and Bury grabbed the lead with their next opportunity. Lowe opened the defence with a ball across the box and although JONES rejected the chance to shoot first time, he brought the ball back to his left before firing into the top corner.
Closer to half-time the Blades saw more of the ball but without troubling the Bury backline, although it was noteworthy that referee Attwell spoke to home skipper Cameron before half-time due to his side's timewasting tactics.
Sharp replaced Calvert-Lewin at the start of the second half and United began with some impetus. Riley kicked off the line from Adams after Lawlor could only parry a Campbell-Ryce cross and then Basham failed to get a clean connection to a Baxter free-kick.
On 54 minutes a patient Bury move saw Long have to push an effort from Mayor around a post and the goalkeeper was called upon again soon after to parry away a powerful header from Peter Clarke.
United's majority share of the possession in the second half failed to pick holes in the well-drilled Bury defence and Flynn and Done were introduced to add fresh legs. As in the first half, the home side happily slowed down the play at every opportunity and the last ten minutes approached with the 839 Blades fans inside Gigg Lane looking for something to cheer.
Basham's header from Baxter's free-kick was wide and Done's ball across the box worried Lawlor. Late on Brayford's header back across goal caused problems but Peter Clarke blocked from Done and three minutes of added time failed to produce another chance.
Bury: Lawlor, Riley, Tutte, P Clarke, L Clarke, Mayor, Jones, Lowe (Pope 80), Soares, Pugh, Cameron.
Unused: Ruddy, Hussey, Mellis, O'Sullivan, Miller, Brown.
Blades: Long, Brayford, Basham, Baxter, McEveley, Campbell-Ryce (Done 70), Reed (Flynn 58), Calvert-Lewin (Sharp 45), Adams, Edgar, Hammond.
Unused: Howard, Coutts, Woolford, McGahey.