The Bees wasted a host of chances and had goalkeeper Simon Moore to thank for earning them a share of the spoils as Graham Turner’s side threatened a smash-and-grab victory.
Shrews striker Marvin Morgan was a constant threat and twice fired wide when well placed before substitute Terry Gornell forced a reflex near-post save with a smart header.
The closest the hosts came to breaking the stalemate was when Clayton Donaldson fired a first-half effort against the outside of a post after a superb Adam Forshaw throughball.
Substitute Harry Forrester almost settled it late on with a curling cross which drifted just wide before Uwe Rosler’s side survived a late penalty scare.
Defender Tony Craig was adjudged to have handled in the box and the referee pointed to the spot, only to reverse the decision after consulting his assistant. Seconds later Brentford’s Paul Hayes had the ball in the net after a Forrester cross, but was denied by the offside flag.