Steve Kabba was outstanding up front for Neil Warnock's side and capped his performance with the third and decisive goal.
Veteran Neil Shipperley's 15th minute strike was hardly a surprise after the Blades had swarmed all over Crewe early on.
Kabba, who scored twice in Alex's 4-0 thumping at Bramall Lane last March proved difficult to pin down and when he pulled the ball back for David Unsworth, Crewe's on-loan keeper Ross Turnbull made a good save with his legs to keep out the veteran defender's low shot.
Turnbull kept Crewe in it denying Kabba after he had got through on goal via a give-and-go with Shipperley.
He also held onto a low drive from Phil Jagielka, which was heading for the bottom corner, at the second attempt, but could do nothing when Shipperley finished off with a rising first-time shot after Jagielka had flicked on Keith Gillespie's corner.
But within a minute an uncharacteristic slip from Paddy Kenny let in Steve Foster for the equaliser a minute later. The experienced keeper allowed Kenny Lunt's free-kick slip from his hands and Foster, returning to the back four for his first start of the campaign, had the simple task of nodding into an empty net from close range.
The goal sparked Crewe into life and Mike Higdon stretched for Steve Jones' cross to head wide at the far post, but Gillespie posed danger when he came close to curling one in at the far post with a classy drive with the outside of his boot.
Crewe's rearguard action suffered a setback when Adie Moses was left writhing in agony after collapsing awkwardly from a challenge with Kabba by the touchline. The Yorkshireman may have suffered a broken leg and he was replaced by Richard Walker.
But Kabba got behind the home defence again before the first 45 closed and his diagonal run just before the break was timed perfectly to take Alan Quinn's pass and it was only the striker's finish which eluded him as he screwed the ball past the far post.
The Blades cut through their hosts' defence twice inside the first ten minutes of the second half to seal the three points. First Quinn fired through Turnbull's legs after Darren Moss allowed him to ghost past him to take Kabba's flick-on.
Then Kabba had a ridiculous amount of time on the edge of the six-yard box to chest down Jagielka's header after Quinn had delivered the ball into the far post. Turnbull could do nothing as the Londoner volleyed home.
Crewe were devoid of ideas as the game petered out, although Slovakian Pavel Suhaj showed good awareness to set up Steve Jones, but Kenny made a fine double save to make up for his earlier blunder.