The Blades had been briefly knocked off their perch by Luton earlier in the afternoon but they had no trouble in seeing off Alex, who slipped to their second league defeat in three days.
The visitors dominated from start to finish but needed second-half goals by Alan Quinn and Steve Kabba - his fourth goal this term - to earn their latest win.
A rare Paddy Kenny error in the 16th minute had gifted Crewe the opportunity of snatching an unlikely point.
Stephen Foster had been the grateful recipient of the Republic of Ireland international's fumble, a goal which had cancelled out Neil Shipperley's opener minutes earlier.
But it was a matter of when, rather than if, United would reclaim their lead and they sauntered to success in a somewhat turgid and low-key second half.
Neil Warnock's men, who claimed their fifth win of this fledgling league season, began in menacing fashion with Phil Jagielka, Keith Gillespie, David Unsworth and Kabba testing Crewe goalkeeper Ross Turnbull in the opening 10 minutes.
With Crewe's defence at sixes and sevens, it was only a matter of time before the Blades scored and the opening goal duly arrived in the 14th minute.
Jagielka flicked on a right-wing corner and Shipperley was on hand at the far post to bundle the ball home.
Amazingly, given United's early dominance, Crewe were level within two minutes.
Kenny Lunt sent in a textbook free-kick delivery which was fumbled by Kenny. Defender Foster was on hand to nod into an empty net from two yards out.
Alex had found their feet but United - displaying their customary street-wise guile - still looked the more dangerous.
Nick Montgomery hit the side-netting from 10 yards out while Kabba, who was being allowed plenty of space in behind the Crewe rearguard, slashed wide with just Turnbull to beat.
Veteran defender Craig Short then headed wide from six yards out as Crewe somehow managed to go into the break on level terms.
It took four minutes of the second half for the Yorkshire club to regain the lead.
Quinn, who was ebullient all afternoon on the left wing, was the gleeful recipient of a Kabba flick-on and he scampered 30 yards - and away from two lethargic defenders - before neatly slotting past a stranded Turnbull.
An increasingly bedraggled Crewe went further behind in the 55th minute when Jagielka nodded a Quinn centre back across goal and Kabba had time to chest down and power home from seven yards out.
Crewe huffed and puffed in the final half-hour, with David Vaughan showing neat skill before sending a 25-yard shot over the bar and Steve Jones saw his close-range effort well saved by Kenny.
But this was another day where Dario Gradi's slight side were outmuscled and overpowered by a more robust and well-oiled Championship rival.