Hartlepool's fourth successive league victory moved them into a play-off place as Bury's poor run continued.
Nolberto Solano opened the scoring with a low finish after a fine run, before Colin Nish prodded home from 10 yards.
Nish headed a Solano cross against the underside of the bar but scored his second with a close-range finish after stealing the ball from Andrai Jones.
Bury, who have not won since selling top scorer Ryan Lowe, had a chance of a consolation but Joe Skarz headed over.
The Shakers are now without a win in seven matches, in all competitions, having not triumphed in a month since the 2-0 victory at Wycombe on 20 August.
They are now just a point above the drop zone in 20th, while Pools climb to fifth.
Bury manager Richie Barker told BBC Radio Manchester:
"We knew what we were going to get from them and it's a big learning curve for two young centre-halves.
"We're not going to panic. We're going to carry on playing the way we want to play as people have had worst spells than this and gone on and get promoted.
"Our aim is to retain League One status and that is not lost just in the space of September."