Pools went into the game with just one defeat in eight but were second best throughout as the Shakers continued their improvement under new boss David Flitcroft, securing back-to-back victories for the first time since November 2012.
Pools midfielder Bradley Walker had a second-minute shot cleared off the goal-line by Frederic Veseli but the visitors soon took control.
Danny Mayor fired the Shakers into the lead after five minutes, his low shot from 20 yards deceiving goalkeeper Scott Flinders, who could only help it into the net.
It was two-nil on 12 minutes when Daniel Nardiello latched onto a slide-rule pass to curl his finish beyond Flinders from the edge of the area.
Visiting 'keeper Brian Jensen saved well from a deflected James Poole shot from long-range as half-time approached.
And the Shakers clinched a comprehensive win after 75 minutes, defender Richard Hinds sweeping in from close range after a long Veseli free-kick landed in the penalty area to lift Bury five points clear of the drop zone.