A late Billy Sharp goal - assisted by two Academy products - earned United the points in an entertaining Good Friday encounter.
Che Adams and Ryan Flynn gave United a two-goal advantage but David Fox and Harry Davis brought Crewe level.
However, Sharp's 17th of the season, following excellent deliveries from Ben Whiteman and Louis Reed gave United the win.
Adkins rang the changes for the Good Friday clash, four in total, with three enforced. Edgar missed out because of international duty, whilst Baptiste and Cuvelier were unavailable through injury. Sammon dropped to the bench, where he was joined by five teenagers, all Academy graduates, including 18-year-old midfielder Brooks. McEveley, McGahey, Coutts and Adams all started.
United were the quicker of the two teams out of the blocks and but for a fine save would have had the lead after just 90 seconds! Done's corner from the right saw Brayford rise to win a back post header but Garratt's strong right arm kept the ball out of the net.
For Crewe, a rehearsed corner routine saw Kirk shoot over the top but the visitors were operating very deep as United continued the early momentum. A good move on the right saw a low Coutts cross met by Sharp but Davis blocked with his face and then, on 15 minutes, the leading goalscorer curled against a post after Adams had chased a loose ball.
United deservedly grabbed the lead, though, on 18 minutes, the result of Basham coming out of a challenge with Fox with the ball. Sharp found Flynn, who laid off to ADAMS, in space, who curled around the exposed Garratt from 15 yards.
Aside from a weak penalty appeal from Jones, Crewe offered little as the Blades dominated the first half. Coutts shot wide after a break out of defence by McGahey and Sharp was adjudged offside as he converted the rebound following a good move which saw Adams force a save out of Garratt.
As half-time approached United continued on the front foot but the only further chance was Adams shooting wide after linking up with Sharp.
United made a nervy start to the second half and the visitors chalked up a number of early chances. Inman beat Basham for pace but his effort was blocked by an accumulation of Long and McEveley, whilst the resulting corner was not cleared and the Blades goalkeeper had to parry away Kirk's shot.
On 51 minutes Bingham was afforded the space to shoot from long range but Long was well positioned to save and Fox's free-kick was deflected over.
Crewe's disappointment at not drawing themselves level was compounded on 61 minutes when the Blades doubled the lead with the first serious attempt of the second half. From a deep Coutts cross, Sharp headed back across goal and after Guthrie failed to clear, FLYNN netted in front of the Kop with an acrobatic effort.
The visitors, though, pulled a goal back with their next opportunity. Another corner routine saw FOX escape Done's challenge before firing into the bottom corner from 25 yards.
Whiteman, Calvert-Lewin and Reed were all introduced inside the last 20 minutes but now Crewe were on the front foot. McEveley cut out a Bakayogo cross but the visitors dragged themselves level on 82 minutes. Ainley got around the back of the defence on the right and his pull-back saw DAVIS beat Long from close range.
There was a twist in the tale, however, with United claiming a later winner thanks to a link-up between three homegrown players. Whiteman's ball down the right channel saw Reed race on and deliver a perfect cross for SHARP to net with a diving header at the Kop end.
The remaining moments, plus six of added time, naturally, saw Crewe searching for a leveller. Kirk hit a half-volley wide and Inman's shot was well held by Long. The final chance was the result of Bakayogo's deep cross but Turton's shot was blocked at the back post by Done and United held on for three more points.
Blades: Long, Brayford, Basham, Flynn, Coutts (Reed 80), Sharp, Done, McEveley, Adams (Calvert-Lewin 78), McGahey, Hammond (Whiteman 70).
Unused: Howard, Sammon, Kelly, Brooks.
Crewe: Garratt, Turton, Guthrie, Davis, Nugent (Bingham 45), Inman, Fox, Jones (Ainley 75), Kirk, Hitchcock (Dalla Valle 67), Bakayogo.
Unused: Ray, Atkinson, Cooper, Richards.