Reds suffered their first defeat in three trips to Bramall Lane after a superb goal by Jose Baxter settled a tight Sky Bet League One encounter early in the second half.
The midfielder curled a superb side-footer into the top corner from 22 yards after Reds had looked the more threatening side in the first half.
Gwion Edwards hit the bar and Bobson Bawling’s follow-up was cleared off the line but United had a great spell immediately after scoring when Michael Doyle and Michael Higdon both went close.
Reds probed for an equaliser and Dean Leacock saw a diving header deflected wide while appeals for a penalty after Izale McLeod was sent spinning to the deck and had a volley blocked in the closing stages were turned away.
Crawley deserved something for their efforts against a side most consider strong promotion candidates but on this evidence there was little to separate the sides apart from Baxter’s moment of quality.
John Gregory made two changes for Reds’ third game of the season against Yorkshire opposition. There were recalls for Conor Henderson and Lewis Young with Gavin Tomlin and Sonny Bradley dropping to the bench.
As they did at Barnsley in their previous away game, Crawley set themselves up to be nice and compact when they didn’t have the ball and to try and hit the Blades on the break.
The hosts were looking to build on an opening win at Peterborough in midweek and made a positive start but it was Mark Howard who was called to make the first save on 14 minutes. Izale McLeod found Bobson Bawling and his angled shot from 12 yards was pushed around the post by the United keeper.
Either side of that both Jimmy Smith and Conor Henderson were off target with strikes from distance then McLeod was left bemused by referee Andy D’Urso’s decision to wave play on when he got goal side of Harrison McGahey before tumbling inside the box.
The hosts looked neat and tidy as they probed for an opening. Jose Baxter fired into the side netting when a corner dropped to him at an extremely narrow angle and on 35 minutes Ben Davies’s curling free kick was bravely held by Brian Jensen with the boots of two opponents flying around his ears.
The home fans were getting restless at their side’s lack of punch and their mood nearly worsened as Reds twice went desperately close to going in front just before the break.
Gwion Edwards, who had switched to the right flank after starting on the left, cut inside and struck a powerful shot that hit the bar and came out to Bawling, whose shot was hacked off the line by Jay McEveley.
Moments later Henderson’s free kick picked out Leacock, whose low cross from the left found McLeod unmarked but the angle was just too tight and he skied his shot.
There were a few boos at the end of a half which had seen bookings for Henderson (foul) and Chris Basham (ungentlemanly conduct).
The mood was to change quickly with United going in front two minutes into the second half. BAXTER received possession 22 yards out and struck a right-foot shot with the side of his foot into the top corner. Jensen’s reactions suggested he thought it might have been going wide but there was probably little the Dane could have done as it flew into the top corner.
It was nearly two a couple of minutes later. Murphy went on a great run which only ended when Lanre Oyebanjo stuck his foot out to dispossess him, but the ball fell to Michael Doyle whose curling effort flew a yard wide of the far post.
United were now dominating and it needed an interception by Joe Walsh to prevent Michael Higdon from firing home after Reds had been prised open down their left-hand side.
It took a few minutes for Crawley to re-group and work their way back into the contest and as the game entered the final third there was still plenty of optimism that they could get back on level terms.
But United came again and Leacock was felled by a drive from McEveley and conceded a corner while the home fans behind the goal howled in unison when Young gave the ball away inside the box before winning it back off Murphy, who was bearing down on goal.
Reds made their first change with 15 minutes to go when Young was replaced by Gavin Tomlin and with the home side seemingly happy to sit on their advantage and try and counter-attack the visitors pushed on in search of an equaliser.
Leacock went close with eight minutes left when he met Henderson’s corner with a diving header which was deflected wide and the Gaffer then brought on Charles Banya for Oyebanjo in another attacking switch. Deep into stoppage time, McLeod's volley from a half-cleared corner, which Jensen had gone up for, appeared to hit a United arm but referee D'Urso again waved away his protests.
United saw the game out and almost got a second on the break through Davies, who side-footed wide with only Jensen to beat, leaving Reds to lick their wounds and go again at Norwich City on Tuesday in the second round of the Capital One Cup.