A disastrous ten-minute first-half spell set the scene for a disappointing night for the Blades.
Despite Conor Sammon giving the Blades the lead, Shrewsbury scored twice in 90 seconds and added a third soon after.
Dean Hammond netted his first for the Club but the Blades, despite second half pressure, were unable to redress the balance.
Reed and Sammon were handed starting berths, replacing Coutts and Campbell-Ryce, who dropped to the bench and were joined by Edgar and Woolford. Scougall missed out with a slight hamstring problem.
After a quiet start, the Blades were indebted to Harris, who made two clearances from consecutive corners on 10 minutes, batting away from Whitbread and then Vernon's deflected shot.
United's first opportunity presented two chances. Good work from Freeman saw Baxter's shot blocked and the rebound from Sharp also crowded out. The Blades did, however, open the scoring with the next move.
With 17 minutes gone, Baxter was stronger than Vernon in midfield and his pass was perfect for SAMMON to shoot low, first time, which found the target off the inside of a post.
It was, though, the start of a sticky period for the Blades, which saw Shrewsbury level and then take the lead inside 90 seconds. Both were results of United failing to clear; KAIKAI picked up a loose ball on the left and cut inside before firing into the top corner and then BLACK fired left footed into the bottom corner after his initial effort was blocked by Hammond.
The home supporters were stunned but worse was to follow. On 35 minutes, Brown beat Baxter down the left and his near post cross was diverted home by the unmarked VERNON.
Before the break there were chances at both ends. Hammond's shot was parried by Leutwiler, but Sammon headed the rebound wide, then Grandison fired over from a Shrewsbury corner and then, on the stroke of half-time, Sharp was off target, under pressure, after linking up with Sammon.
Adkins reacted by making a triple change at the break. Baxter, Collins and Reed were replaced by Done, McEveley and Edgar, with the latter two making a central defensive pairing.
Blades fans thought a penalty should have been awarded on 47 minutes when Whitbread handled as he challenged Sammon. Referee Clark awarded a free-kick on the edge of the box, but after a direct shot from Harris struck the wall, Shrewsbury broke and an unmarked COLLINS fired across Long and into the corner of the net.
Black fired just wide from distance as United remained shocked. Sammon's shot was blocked and Done could not control the rebound as the home side looked for a route back into the game. The arrears were reduced on 65 minutes when an inswinging Harris free-kick was glanced home at the back post by HAMMOND.
United's chances continued. From a corner, Sharp headed back into the danger area but Done could not convert and then Freeman's swerving long-range blast was saved by the body of goalkeeper Leutwiler. From the next corner Edgar headed over.
Late on Harris curled a free-kick over the top from a good position and Adams could not keep his shot down following another attack.
In injury-time, from Sharp's cross, Edgar headed down and Leutwiler saved from Basham from close range and Freeman missed from the middle of the box to sum up United's luck.
Blades: Long, Freeman, Harris, Basham, Sharp, Baxter, Collins, Sammon, Reed, Adams, Hammond.
Unused: Howard, Coutts, Done, Woolford, McEveley, Campbell-Ryce, Edgar.
Shrewsbury: Leutwiler, Whitbread, Grandison, Black (Lawrence 79), Collins (Barnett 79), Kaikai, Brown, Gerrard, Vernon, Ogogo, Cole (Akpa Akpro 76).
Unused: Whalley, McAlinden, Burton, Smith.