The Shrimps forced the pace from the first moments with Ryan Williams fizzing a shot from 25 yards that was deflected wide by a desperate intervention from Jim McNulty.
Kevin Ellison then went close when he forced goalkeeper Brian Jensen into a sharp save at his near post with a shot from an acute angle.
Jensen had to be alert again on the stroke of half-time when he showed good footwork to tip over a dipping Andy Fleming shot.
The Shakers goalkeeper, who will leave the JD Stadium this summer, produced a point-blank stop from Fleming in the second half too to preserve his clean sheet.
Bury’s only effort of note came late in the second half when Danny Mayor missed the target when he should have converted Craig Jones’s right-wing cross.
Morecambe defender Ryan Edwards had the final chance of the game but volleyed over from six yards out.