Two goals from Liverpool loanee Conor Coady condemned Reds to their first home league defeat of 2014.
The midfielder struck on 18 and 63 minutes as an impressive Sheffield United side made it 12 wins from their last 14 games in league and cup.
Crawley had their chances with Mark Howard making good saves from ex-Blade Kyle McFadzean and Andy Drury twice, but there was no way through a well-organised visiting defence.
Boss John Gregory made two changes to the side beaten at Oldham on Saturday. There was a first League start since December 29 for Sergio Torres with Dannie Bulman returning alongside the Argentinian in midfield. Josh Simpson (hamstring) and Sullay Kaikai dropped out with Andy Drury taking the captain’s armband.
Reds began well and United keeper Mark Howard was forced to make a good save after three minutes when McFadzean flicked on Billy Clarke’s free kick and Howard palmed it around the post at full stretch.
The hosts definitely settled first and were putting together some decent moves. McFadzean met Drury’s corner but headed wide then the skipper threaded a pass into Matt Tubbs’ path but Howard came out to the edge of his box to smother.
So it was a surprise when the Blades cut through to open the scoring on 18 minutes. Mark Connolly and Mike Jones both fell trying to stop Jamie Murphy down the left and although McFadzean managed to block his cross the ball fell to Conor COADY 12 yards out and he smashed it past Paul Jones.
Reds lost their way for a while but they nearly levelled on 27 minutes. Clarke tapped a free kick to Drury whose low drive hit the side netting.
Against a side with such a strong defensive record Reds now needed to show patience in the search for a way through, the manager urging them to ‘pass it’ from his technical area.
Four minutes before the break the visitors only cleared a corner as far as Connolly who blasted his right-foot shot just over from 15 yards.
United began the second half on the front foot but Reds should have levelled on 54 minutes. Mike Jones’ shot ricocheted into the path of Drury who went for goal from a tight angle.
Howard blocked his shot with an outstretched leg and Tubbs could only force the rebound over the bar.
It was not hard to see why the visitors have such an excellent defensive record. They rarely stood on ceremony with the highly rated Harry Maguire showing why he is being coveted by some top clubs.
It got worse for Reds on 62 minutes. Murphy got past Connolly down the left and his cross found its way to COADY 15 yards out whose powerful drive flew past a helpless Jones.
Reds made their first change on 65 minutes when Jamie Proctor replaced Torres as the hosts switched to a 4-4-2 in their search for a way back into the game.
Neill Collins nearly put through his own goal on 73, sliding in as his keeper came out to clear with the ball rolling wide of the post. From the corner Proctor’s shot produced another fine save from Howard who pushed it away for another corner that McFadzean met but headed wide.
Rory Fallon replaced Tubbs with 11 minutes to go but it was United who were denied a third goal when Jones pulled off a terrific save to keep out Murphy’s drive and United saw the game out comfortably enough to stretch their unbeaten away run to six games.