March settled the issue dramatically by heading in an assist by Joel Tabiner at the far post, Crawley having gone down to nine men 25 minutes from time.
Crawley's former Arsenal and Sunderland keeper, Vito Mannone, twice required treatment in the opening 18 minutes after heavy collisions as Crewe edged the opening exchanges.
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Crewe keeper Tom Booth, standing in for the injured Ian Lawlor, dived to save a low shot from Louis Watson before the break.
Crawley were hit by a sending off six minutes after the break when Kellan Gordon received a straight red card for violent conduct in an off-the-ball incident involving midfielder Joe White.
The Red Devils then went down to nine men after 65 minutes when Malcolm Ebiowei also saw red on his Crawley league debut for throwing the ball at Calum Agius.
Referee Andrew Humphries also gave Crewe captain Mickey Demetrou his marching orders late on for a second bookable offence in a game which had a total of 18 minutes of added time.
