Ryan Lowe tested Shrimps goalkeeper Barry Roche early on while the visitors responded with Mark Duffy seeing his curling shot well saved by Wayne Brown.
Brown came to Bury's rescue again three minutes before the break when he stuck out an arm to meet a close-range shot from Paul Mullin.
The game opened up after the restart and Morecambe's Craig Stanley twice went close before the hour mark. Bury's Mike Jones was then thwarted by a super save from Roche, before the visitors came back again with Garry Hunter seeing a firm drive deflected into the side-netting.
The Shakers went closest to nicking a victory late on as Lowe headed inches wide and then skipper Stephen Dawson blazed over after racing clear.