Shirtliff took control at Gigg Lane following Richie Barker's departure to Crawley, but could not inspire his side to victory in their npower League One opener.
Chances were at a minimum during the first half, although Sam Saunders came close with an improvised header from Shaleum Logan's cross and Shakers goalkeeper Trevor Carson also needed his wits about him to save an Adam Forshaw shot.
Bury rarely threatened, but David Worrall saw a stinging effort blocked by Scott Barron before a wildly-deflected Steve Schumacher shot crept narrowly wide.
Brentford made all the early running after the break and Adam Lockwood produced a timely interception to deny Leon Legge a chance at point-blank range.
Substitute Mark Cullen blazed a good chance wide for Bury and Worrall had a strong penalty appeal denied following a challenge from Harlee Dean inside the box before Clayton Donaldson wasted a glorious chance in injury time, firing into the side-netting when clean through.