The Shakers striker pounced in the 42nd minute to secure a richly-deserved victory at Gigg Lane.
Bury almost went ahead in the 21st minute when Ben Futcher crossed into the box and Paul Scott headed agonisingly wide.
The Shakers did eventually break the deadlock, Glynn Hurst and Michael Jones combining to find Bishop, who cut in from the right to fire home left-footed from 16 yards.
Nathan Elder headed over for Brentford on the hour mark but Hurst should have sealed it with 20 minutes remaining when he struck a post from Elliott Bennett’s through-ball.
Wayne Brown had to be alert to save a 25-yard drive from Alan Connell shortly after, while Kevin O’Connor blazed over for Brentford at the death.