Academy coaches Mark Trueman and Conor Sellars have been placed in temporary charge while the club embark on the process of finding their seventh manager in three years.
And Bradford got off to a good start for their caretaker bosses by taking the lead on 11 minutes.
Austin Samuels, playing in a wide-right role, ran in behind the home defence to collect Connor Wood’s long ball.
Crawley claimed offside but the flag stayed down as his cross was mishandled by goalkeeper Glenn Morris, allowing Lee Novak to score from the rebound.
But Bradford have struggled to keep clean sheets with only two in the league this season.
And they shot themselves in the foot after 26 minutes, Harry Pritchard heading into his own net from Sam Matthews’s free-kick.
Instead, it was the visitors who posed the attacking threat – and but for Morris, would have returned north with a second away win of the season.
The goalkeeper made three excellent saves in the second half from Samuels, Pritchard and Novak.
Crawley: Morris, Davies, Craig, Tunnicliffe, Dallison (Nadesan 62min), Tsaroulla, Powell, Hessenthaler, Matthews (Ferguson 62min), Watters, Nichols (Ashford 80min). Unused substitutes: Nelson, McNerney, Sesay, Galach.
Bradford City: O’Donnell, Cousin-Dawson, P O’Connor, A O’Connor, Wood, Sutton, Watt, Samuels (Scales 69min), Cooke, Pritchard (French 78min), Novak (Donaldson 78min). Unused substitutes: Longridge, Sikora, Shanks, Hornby.
Referee: S Purkiss (London).
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