It was a fourth successive league win for Uwe Rosler's side, while Crawley have now lost their last three games.
Brentford, who completed the double over Crawley last season, made the perfect start when midfielder Forshaw gave them the lead after only five minutes from the penalty spot.
Forshaw sent keeper Paul Jones the wrong way after defender Kyle McFadzean had brought down Kadeem Harris in the area.
Former Everton man Forshaw later shot over from the edge of the area before Crawley midfielder Andy Drury tested visiting keeper David Button with a low shot through a crowd of players.
Italian striker Marcello Trotta threatened for Brentford shortly after the break with a shot from an acute angle, which was saved by Jones.
Drury put a diving header wide at the other end after a Mike Jones pass had set him up in front of goal.
Brentford striker Clayton Donaldson shot wastefully wide after being put through by Forshaw and Trotta was also off target having got himself into a good position.