Ryan Lowe twice went close for the home side as they dominated the opening stages of the game.
Full-back Craig Jones gave the Shakers the lead with a curling strike from 20 yards out.
Joe Riley denied the Blades after the break with a fine clearance off the line before a last-ditch block denied John Brayford a late equaliser.
Bury registered their first league victory in four games with a win over Sheffield United at Gigg Lane.