Jose Baxter's goal early in the second half helped the Blades to a first home league win of the campaign.
The striker beautifully placed home to secure three more deserved points for United against Crawley.
In a hard-fought game, the Blades were professional in the approach to secure victory.
Following the midweek win at Peterborough, boss Clough made one change to the starting line-up as Higdon took over from McNulty, who dropped to the bench. The visitors included one-time former Blades loanee McLeod.
United persisted with the 4-4-2 formation, which gleaned three points at Peterborough, and were allowed plenty of the ball in the opening five minutes, which resulted in Higdon heading over a Davies corner.
Following the initial spell of Blades pressure, Smith fired wide for Crawley before a good opportunity materialised on 14 minutes. A mix-up between Collins and Doyle presented Young with a shooting chance but Howard was down to his right smartly to push around a post.
The opening half hour's action saw Crawley content to operate on the counter-attack and with pace on their side they looked dangerous. Collins missed a header, exposing McGahey, but referee D'Urso showed little interest when the young centre half and McLeod clashed on the edge of the box.
It was hoped that a spell of rain and Murphy and Baxter switching positions would improve United's chances of opening the scoring but Crawley were protecting their box well. On 37 minutes a foul by Edwards on the edge of the box resulted in Davies whipping in a free-kick but Jensen gathered at the second attempt, despite being under pressure from Murphy and Alcock.
Henderson and Basham picked up bookings for separate fouls as half-time approached but the away team finished the half the stronger. On 42 minutes Edwards found space in midfield and shot at the crossbar, with Howard picking himself up it was left to McEveley to block the rebound from Bawling. Another chance saw Leacock keep the ball in and McLeod fire into the Kop with the Blades defence asking questions of each other.
The Blades made the perfect start to the second half, taking the lead on 48 minutes. A patient move saw the ball shipped out to the right and after Alcock's lay-off, BAXTER set himself on the edge of the box before curling into the corner of the net, leaving goalkeeper Jensen stranded.
United had chances to extend the lead on a couple of occasions straight after the goal. Following a driving Murphy run, Baxter curled just wide of an upright and then Higdon's effort was blocked by a defender after good work between Murphy and Davies on the right.
The chances continued to fall for the Blades. Goalkeeper Jensen pushed away from Basham after a low cross from Davies and McEveley's long range effort was blocked.
Flynn replaced Baxter for the final 20 minutes, leaving Murphy in attack with Higdon, whilst the next chance dropped to McGahey after a Davies corner but the youngster could not find the target.
Reed replaced Higdon, suggesting a change in formation for the final ten minutes, whilst Crawley's first serious chance of the second half arrived on 83 minutes from Henderson's corner but McGahey did enough to put off Leacock.
McEveley limped off late on, replaced by Harris, and United were professional in protecting the lead, plenty of yards from Howard's goal. In injury-time the Blades had a chance to double the lead when Flynn and Harris combined on the left but this time Davies, in space in the centre of the goal, shot wide.
Leacock was added to Crawley's attack but United held firm at the back and kept possession well - until the fourth and final minute of added time. Goalkeeper Jensen was up for a corner and challenged with his opposite number but Howard punched clear and McLeod's late shot was blocked by a barrage of bodies as the Blades held on.
Blades: Howard, Alcock, Basham, Doyle, Murphy, Baxter (Flynn 69), Collins, Davies, McGahey, McEveley (Harris 86), Higdon (Reed 79).
Unused: McNulty, McGinn, Campbell-Ryce, Turner.
Crawley: Jensen, Oyebajo (Banya 84), Dickson, Henderson, Leacock, Edwards, Smith, McLeod, Walsh, Young (Tomlin 75), Bawling.
Unused: Bradley, Rose, O'Connor.