Defender Danny Holmes gave unbeaten Rovers the perfect start with the lead in only the second minute with his first goal for the club, firing an angled shot from 25 yards into the bottom corner.
Midfielder Josh Simpson levelled after 12 minutes with an unstoppable right-footed shot from 20 yards after good work from Nicky Ajose, and defender Kyle McFadzean (16) put Crawley ahead by firing into the roof of the net after Nicky Adams' corner was not cleared.
Rovers goalkeeper Owain Fon Williams saved shots from Gary Alexander and Mike Jones before Welshman Cassidy, on loan from Wolves, made it 2-2 by heading in at the far post after 40 minutes following a 70-yard run by Zoumana Bakayogo.
French striker Jean-Louis Akpa Akpro restored Rovers' advantage a minute after the restart by running through a hesitant defence to beat stranded goalkeeper Paul Jones, and Cassidy (54) seized on a mistake by Mark Connolly to make it 4-2.
Former Swansea and Leeds midfielder Andy Robinson sealed Rovers' comfortable win in the last minute by netting his seventh of the season from close range.