Bury dominated the first half and got a dream start through Jones' seventh goal of the season. Ryan Lowe was fouled just outside the box and Jones clipped a right-foot effort into the top corner.
The hosts remained on top with Dave Worrall heading over from a Stephen Dawson cross before skipper Dawson dragged an effort wide.
Jones tested Adam Bartlett just after the break before Sam Gwynne scrambled back to clear after Lowe's shot beat Bartlett.
The visitors began to peg Bury back as Leon Constantine fired wide from 12 yards before being denied by Wayne Brown. And the Bury keeper produced a fine late save to tip over a Constantine header and secure victory.