Two goals from Harry Maguire helped get Nigel Clough's reign as Blades boss off to a winning start.
Maguire netted his third and fourth goals of the season in the first half as United took the game to Crewe.
Ryan Flynn added a third after the break before Max Clayton netted a consolation for Crewe.
New boss Clough made two changes from the team that drew at Peterborough in midweek. Hill and King were recalled, with Coady dropping to the bench and Brandy ruled out with a virus.
United began with a 4-4-2 formation with Lappin ahead of Hill on the left. The opening ten minutes were quiet in terms of chances, Oliver headed wide for the visitors, whilst Doyle fired over the top after Maguire had taken down McGinn's corner.
Following a foul by McMahon, Osman's header from Moore's free-kick needed turning over the crossbar by Long before the Blades scored the first goal in Clough's reign on 18 minutes. Mellor committed a foul on McGinn, who recovered to curl in a free-kick from the left, which MAGUIRE swooped to head home at the back post.
Taylor was denied a second by a fine save from Phillips after Flynn and McGinn linked up but the Blades doubled the lead just before the half hour mark and Crewe did not learn their lesson. The same two Unitedites linked up, this time McGinn delivered a corner from the left flank and MAGUIRE was ideally placed at the back post to nod home.
With Taylor and King working hard in attack, United were causing the visitors all sorts of problems, but between the half hour and the referee's half-time whistle neither goalkeeper was seriously tested and it was the home fans who were the happier as the players headed off for a break.
Early in the second half Crewe had a great chance to reduce the arrears but Moore could not direct Inman's cross with the outside of his right boot and the visitors were soon three down. Maguire's tough tackle in midfield enabled Lappin to play in FLYNN, who cut in from the left and shot low, beating Phillips at his near post.
Blades chances continued. Taylor sliced an effort wide and then flicked on a Lappin cross for King, but he was denied by goalkeeper Phillips, racing off his line. Moments later King was replaced by Miller, a former Crewe Academy graduate.
Phillips collected efforts from Collins and Miller, whilst Crewe went close on 76 minutes - a Moore corner was won by Ray and Aneke's volley clipped the top of the crossbar. The visiting side's substitutes then combined to give the travelling fans a little hope as Aneke played through the middle of the defence and CLAYTON poked beyond Long.
In the final ten minutes Coady replaced the hard-working Taylor but Crewe were now the more offensive outfit. Grant's long-ranger was at Long and then a free-kick from Aneke struck the wall.
Cuvelier took over from Lappin as four minutes of injury-time approached and the Blades saw out the remaining moments with Miller as the lone striker, cementing a pleasing start for Clough.
Blades: Long, Maguire, Hill, Flynn, Doyle, Taylor (Coady 82), McGinn, Collins, King (Miller 69), Lappin (Cuvelier 90), McMahon.
Unused: Howard, Westlake, Baxter, Murphy.
Crewe: Phillips, Tootle, Mellor, Dugdale, Osman, Moore, Ray, Inman, Oliver (Clayton 73), Turton (Aneke 57), Grant.
Unused: Ellis, Guthrie, Nolan, Garratt, Molyneux.