But after that the visitors took control. David Worrall wasted Bury's best opening, dragging wide after being released by Andy Morrell.
The Shakers came close again in the 22nd minute when Ryan Lowe collected Michael Jones' quick free-kick and shot against the post.
The game opened up after the interval with both sides looking to attack, Adomah again going close for the Bees.
As the half wore on Bury looked the side most likely to win it and substitute Andy Bishop was thwarted from close range by the impressive Cole, who also parried a late Jones header.