United wasted a host of chances before slipping out of the EFL Cup at Bramall Lane on Tuesday night.
Leon Clarke scored early on but a late goal from Ryan Lowe and another in extra-time gave Crewe victory. The Blades hit the crossbar twice in the extra period but were unable to draw level.
Chief Wilder named an unchanged starting line-up for the encounter, although there were a couple of changes on the bench where Freeman and Calvert-Lewin replaced Wallace and Adams.
United blasted out of the blocks from the kick-off and Crewe were unable to cope. On two minutes, from Clarke's cross, Sharp's header hit Done and then, from a poor goalkeeper kick, Clarke was in the clear but Garratt recovered to claim at the striker's feet.
A poor Turton back-pass then gave Clarke another opportunity and Garratt saved again - but the opening goal arrived on six minutes. Duffy delivered the cross from the left and CLARKE held off his marker to turn home his first Blades goal at the front post.
The Blades continued to dominate. Clarke flashed across goal with the goalkeeper beaten and from Hussey's cross, Fleck arrived at the back post but Garratt saved.
Cooper looked Crewe's major threat and he cut in from the left and shot just wide on 25 minutes. After Clarke's shot was smothered by Garratt from Sharp's lay off, Cooper went close for the visitors again, lifting a shot over on 34 minutes.
Close to half-time there were good chances at both ends. Basham's tackle and another short back pass created half a chance for Clarke, who poked wide under pressure from goalkeeper Garratt, and for Crewe, Cooper delivered an inswinging free-kick and after the ball hit Brayford, Ray slid in but Long saved on the line.
Twice Brayford and Duffy linked up well early in the second half but United were unable to extend the advantage. Duffy was also involved on 57 minutes as Sharp hammered across goal and wide from an acute angle.
Nugent's header denied Clarke a certain second from Done's cross whilst Crewe's Cooper was soon involved again, firing a free-kick into the wall after he was fouled by O'Connell on 65 minutes.
At the other end Nugent went mightily close to levelling for the visitors when he met a Cooper corner, whilst Clarke thought it had claimed a second on 77 minutes but he was adjudged offside as he converted the rebound from substitute Scougall's shot.
United wasted a glorious chance for a second on 87 minutes when Calvert-Lewin won possession but he scuffed wide when in the clear following McNulty's lay off and then the Blades were made to pay.
Long saved a half-volley from Lowe but Crewe levelled from the resulting corner. The Blades wasted several chances to clear before the ball dropped to LOWE to hammer beyond Long with 89 minutes gone, forcing extra-time.
Early in the extra period, Duffy scuffed a shot straight to goalkeeper Garratt and Brayford had a half-volley blocked.
But again United were left to rue missed chances. From a deep Jones free-kick, Nugent won the first header and LOWE lost his marker to side-foot past the exposed Long.
Before the break in extra-time, Calvert-Lewin fired over after a burst down the left channel, whilst just after the restart Fleck smashed an effort against the crossbar before Nugent cleared.
Basham fired over with a half-volley as United tried to force Crewe back and the Blades hit the crossbar for a second time on 115 minutes, Duffy smashing against the woodwork from 25 yards.
It proved to be United's last meaningful chance as the EFL Cup hopes for another season were dashed by the final whistle.
Blades: Long, Brayford, Hussey, Fleck, O'Connell, Basham, Sharp (McNulty 70), Done (Scougall 76), Wilson, Duffy, Clarke (Calvert-Lewin 83).
Unused: Ramsdale, Wright, Coutts, Freeman.
Crewe: Garratt, Turton, Ray, Nugent, Jones, Dagnall, Saunders (Lowe 68), Cooper, Bakayogo, Bingham (Ng 45), Kirk (Kiwomya 59).
Unused: Guthrie, Davis, Richards, Udoh.