Bees' defender Miguel Llera came closest to breaking the deadlock but his first-half header came back off the Wednesday crossbar.
Gary Alexander headed over unmarked in the opening minutes before visiting keeper Richard Lee kept out James O'Connor's acrobatic effort at the other end.
Llera then hit the bar from Sam Saunders' free-kick before Lee had to be alert to twice deny Owls winger Ben Marshall in the closing stages of the first half.
The Bees continued to threaten after the restart and Alexander narrowly failed to connect with Niall McGinn's cross. Owls full-back Lewis Buxton then broke clear down the right but his inviting cross evaded Ryan Lowe before Marshall lost his footing.
Rob Jones was the next to go close for the home side but the defender's header was somehow scrambled wide from inside the Brentford six-yard box. And the Bees held on to extend their unbeaten away record to seven League games.
