Moses Odubajo put Orient in front but by half-time Gary Alexander and Andy Drury had given the visitors a lead before Nicky Adams made it 3-1, only for Orient to scramble an effort from Marvin Bartley to give themselves late hope.
Odubajo calmly slipped the ball past Crawley keeper Paul Jones after 14 minutes to open the scoring, but the visitors hit back with two goals in a seven-minute spell.
Alexander, who spent five years at Orient earlier in his career, met a cross from the right delivered by Mark Connolly to head the ball powerfully into the net from close range.
Controversy surrounded the visitors' second goal after Lloyd James had flung himself into the path of a shot only for referee Trevor Kettle to suggest a handball offence.
Alexander saw his spot kick saved on to the post, but the ball rebounded to Drury for him to score from close range.
With Orient pushing men forward after the interval, Adams capitalised on a mistake by Mathieu Baudry when lobbing the ball over advancing keeper Jamie Jones to give Crawley a two-goal cushion.
Substitute Bartley applied the finishing touch to a free-kick late in the game, but the visitors held on to give new manager John Gregory his first win.