Sam Saunders struck after 73 minutes to give the Bees their second win over Town in four days following Saturday's 2-1 league victory.
"I've always said it's a squad that gets you results and this result proves that," he said. "We had quality players who came in and did a job for us.
"Crawley were one of the stronger sides in the southern section with us so to get past them means we are one step closer to Wembley, although whoever we face will present us with a different challenge."
Rosler, who will find out Brentford's opponents in the southern area quarter-finals when the draw is made on Saturday, added: "Crawley came at us in the second half as we knew they would, but realistically we should have been ahead by half-time."