Albion came into the game second in League Two and had the better of the first half. But the Brewers will be happy with a point as Bury, looking to drag themselves away from the drop zone, pushed for a late winner.
Both teams had chances and Billy Kee should have broken the deadlock but his close-range header was superbly kept out by Shakers goalkeeper Brian Jensen.
Adam McGurk and Jimmy Phillips both tested Jensen from outside the box while at the other end Anton Forrester fired straight at visiting keeper Dean Lyness.
Zeli Ismail then volleyed just over from Phillips' 25th-minute cross before Bury replied with Ashley Grimes shooting too close to Lyness.
McGurk's half-volley fizzed just wide before the break while Phillips' long-range strike was deflected off target.
Forrester was the first to threaten after the restart, firing over from a Grimes throughball, while strike partner Daniel Nardiello hit a tame effort at Lyness.
Phil Edwards then made a last-ditch tackle to deny Nardiello after a fine run by the on-loan Rotherham striker to ensure the scoresheet remained blank.