Neil Warnock's high-fliers still have an eight-point gap over nearest rivals Watford, but the Hornets and Leeds have games in hand.
And it was Crewe who carved out the two best chances, former Blades striker Gareth Taylor and Luke Rodgers hitting the side netting from close range.
The United boss sliced apart the side beaten 3-2 by QPR last week, dropping Paul Ifill, Neill Collins, Rob Kozluk, Nick Montgomery and Geoff Horsfield, with midfielder Michael Tonge sidelined by flu.
And it was Republic of Ireland international Alan Quinn, recalled to the side in the shake up, who almost snatched the lead in the second minute.
Crewe defender Jon Otsemobor headed Keith Gillespie's deep cross straight to Quinn, who pulled his volley inches wide of the target.
Quinn was the provider minutes later, cutting inside before picking out Phil Jagielka whose 20-yard strike on the slippery snow covered pitch flashed the wrong side of the far post, with goalkeeper Ross Turnbull struggling to cover.
Otsemobor cleared from under the nose of his keeper and Jagielka's downward header almost found the incoming Danny Webber, before Quinn was foiled by Turnbull as he drilled in a right-foot effort.
A rare visiting attack saw Algeria international Madjid Bougherra pick out on-loan Nottingham Forest striker Taylor, who ran on unchallenged but could only slam his shot from 12 yards into the side-netting.
Crewe somehow survived as United piled on the pressure. Full-back Chris Armstrong saw his low strike deflected and captain Chris Morgan had a stinging shot blocked and the ball spun over the bar.
Turnbull pulled off a stunning save to foil Webber from close range, Garry Flitcroft shot narrowly wide and substitute Paul Ifill, on for the injured Quinn, had a long-range effort smothered by the busy Turnbull as United failed to break the deadlock.