First-half goals from Sam Saunders and Clayton Donaldson set up only Brentford's second win in the last seven league games before Nicky Adams pulled one back for the home side in the 65th minute.
In a dramatic start, the Red Devils had Irish defender Mark Connolly sent off after just 50 seconds for a professional foul on Donaldson.
Connolly pulled down Donaldson denying him a goalscoring opportunity inside the penalty area, but Donaldson then stepped up to blaze the subsequent penalty wide of goalkeeper Paul Jones' right-hand post.
Brentford broke the deadlock after 16 minutes when midfielder Saunders rifled home from 15 yards after being set up by Donaldson.
Donaldson doubled the advantage in the 31st minute by cashing in on a defensive mix-up between Joe Walsh and Connor Essam to walk the ball into an empty net for his 20th goal of the campaign.
Crawley threatened shortly after the break when Mat Sadler's free-kick was headed over the crossbar by Donaldson before substitute Aiden O'Brien shot narrowly wide.
Brentford were reduced to 10 men after 59 minutes when Adam Forshaw collected a second yellow card for diving.
Winger Adams gave Crawley hope with a superb 30-yard strike into the top corner midway through the second half, but Brentford held on.