Jonathan Douglas grabbed the only goal of the game at Griffin Park after 58 minutes to move Brentford nine points clear of third-placed Rotherham United, who went down 1-0 to Ben Davies' 90th-minute penalty for neighbours Sheffield United.
Davies' dramatic late winner against Rotherham came after Craig Morgan had been penalised for fouling Harry Maguire.
While Brentford appear firmly on course to reach the Championship, Crawley could well be heading in the other direction after a sixth successive defeat.
Crawley are just three points above the relegation zone, although they still have at least one game in hand on the rest of the teams in League One.
Peterborough United strengthened their hold on sixth by beating Gillingham 2-0.
Britt Assombalonga and Tommy Rowe scored either side of half-time for Darren Ferguson's men, who now have a six-point lead over MK Dons.
In League Two, Tyler Harvey and captain Conor Hourihane found the net as Plymouth Argyle won 2-1 at Newport County.
Plymouth are two points behind York City, who occupy the division's final play-off place.
