The Blades had not recorded a win in their previous four games but goals by Chris Armstrong and Jack Lester reignited their quest for promotion.
Armstrong's goal, his first for the club, came against the run of the play in the 20th minute after the visitors had bossed the early exchanges and looked more likely to score.
The Blades' summer signing headed in a punt into the box from Stuart McCall which went in off the post to leave goalkeeper Clayton Ince wrong-footed.
Crewe had made an assured opening showing the confidence which comes from an unbeaten run of five games, while United were struggling with their opponents' passing game.
The Railwaymen had the ball in the net in the 10th minute when an incisive passing move involving Anthony Tonkin and David Vaughan resulted in Steve Jones striking the ball home - but he was adjudged to be marginally offside.
The Blades visibly grew in confidence following the goal and in the 21st minute Ashley Ward tested Ince with a 25-yard drive.
A minute later the same player hit the outside of a post following a pinpoint cross from the left by Armstrong.
The visitors almost levelled through Stephen Foster, whose header was inches wide after an inswinging cross from the right.
Steve Jones was also inches away from equalising after turning and shooting following a pass by Dean Ashton but goalkeeper Paul Gerrard got a touch to turn it around for a corner.
Lester was in the thick of the action as the second half started with the same attacking play from both sides and there were cries for a penalty when he went down under the attention of two defenders.
Shortly after Lester was a whisker away from doubling the lead with a turn and shot following a pass from Armstrong which went just the wrong side of the post.
The former Nottingham Forest striker was almost rewarded for his persistence when a clearance by Ince rebounded off him to go inches over the bar after he charged down a backpass by Tonkin after 61 minutes.
United were firmly in control and in the 73rd minute were awarded a contentious penalty when Neil Sorvel was harshly adjudged to have handled. Lester buried the ball to the goalkeeper's left to convert his second spot-kick in two games and seventh goal of the season.
Although Crewe kept pushing the hosts were dominant and the result was never in doubt as United ended a run of two home league defeats to register three vital points.