John Dempster headed home Scott Davies's corner after the break to equalise before Davies blasted in a stunning 30-yard strike.
Sergio Torres was sent off a reckless lunge on Bees defender Clovis Kamdjo.
VIEWS FROM THE DRESSING ROOM
Crawley caretaker manager Craig Brewster told BBC Sussex:
"You saw what happened with the manager going to Rotherham and me taking over, to get a win today after all that is very pleasing, so I'm delighted for the players and the club.
"Things happened very quickly. That's what happens in football. Players move on, managers move on.
"I'm taking the team until the end of the season, and the players have responded really well. You've seen the support today, I thought it was fantastic, so it gave them a smile and they'll go home happy."
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