sheffield united remain fifth after a late penalty from substitute Marc McNulty rescued a point.
McNulty levelled in the 81st minute after Matty Done had been brought down by Ryan Dickson, Michael Doyle having flicked on goalkeeper Iain Turner’s clearance.
The equaliser left the Blades nine points off the top two following a six-match unbeaten run.
Debutant and former Sheffield Wednesday defender Richard Wood headed the hosts into the lead against the run of play in the 63rd minute, but Crawley dropped to second from bottom despite stretching their unbeaten run to three games.
Their interim manager and former Blades chief Dean Saunders said: “On reflection, you would’ve taken a point before the game but the way we’ve been playing, we were expecting to win. We just had to hang on, so it’s an awful goal we’ve conceded.
Blades manager Nigel Clough said: “Over the course of the game, it was two points dropped.
“But then again when you’re 1-0 down with 20 minutes to go, again, then it’s good to get a point.
“We dominated at times but didn’t quite get a break in the box.”
The Blades had enjoyed 69 per cent of the possession in the first half but had failed to break through.
Crawley took advantage when Anthony Wordsworth swept in an inswinging free-kick towards the penalty spot and Wood, signed on loan from Rotherham, turned the ball home.
McNulty stayed cool to level from the spot and then had a shot turned behind as the Blades sought a winner, which failed to materialise in the five minutes of stoppage time.
Jason Holt, on loan from Hearts, had again been ruled out through injury for the Blades and it remains to be seen if he will be fit for tomorrow’s visit of Peterborough.
Crawley Town: Price, Young, Dickson, Wood, Bradley, Edwards (Ward 90), Wordsworth, Fowler (Smith 58), Tomlin, McLeod, Pogba (Bawling 83). Unused substitutes: Jensen, Morgan, Anderson, O’Connor.
Sheffield United: Turner, Harris, Brayford, McEveley, Kennedy (Freeman 34), Done, Coutts, Doyle, Basham, Flynn (McNulty 74), Murphy. Unused substitutes: Willis, Davies, Reed, Adams, Calvert-Lewin.
Referee: C Breakspear (Surrey).
Man of the match: Richard Wood.