When Monkhouse headed in a Jon Franks cross on 42 minutes against the Shakers, it was Pools' first goal in 762 minutes of football.
They only had to wait another 30 minutes for another, Monkhouse again meeting a Franks cross to head in at the near post.
Pools were never troubled against a desperate Shakers’ side, who dropped to the bottom of League One at Pools' expense.
Simon Walton hit the post on five minutes for Pools, but they had to wait until two minutes before half time to score.
The Shakers then had Craig Fagan sent off for a foul on Darren Holden before half-time and there was still time for Franks to miss an open goal before the break.
Andy Bishop had the visitors’ best effort, holding off Jack Baldwin and curling over on 55 minutes.
But Monkhouse clinched the points on 72 minutes. Franks crossed from the right and Monkhouse’s near post header crept over the line.
HARTLEPOOL UNITED: Flinders; Austin, Collins, Hartley, Holden; Franks, Walton, Baldwin, Monkhouse; Wyke (Humphreys 46), James (Boagey 90). Subs: McHugh (gk), Luscombe, Rowbotham, Rutherford, Hawkins.
BURY: Belford; Worrall, Sodje, Eastham, Ebanks-Landell; Fagan, Thompson (Mezague 77), Carrington, Soares (Regan 78); Bishop, Ajose (Clarke-Harris 55). Subs: Carson (gk), Regan.