An electrical fault meant proceedings got underway 20 minutes later than scheduled, but that didn't stop Dwayne Mattis from making himself the hero with his first goal of the season.
Victory for the Shakers made it six successive wins in domestic competition and lifted them three places up to eighth in the table.
The midfielder's seventh-minute strike was enough to secure the points as Chris Casper's side dominated at Gigg Lane.
Bury started the brighter and went close to taking the lead in the second minute but Andy Bishop's powerful left-foot shot was well-saved by Wrexham keeper Michael Ingham.
But the home side didn't have long to wait for the opener when, four minutes later, David Buchanan crossed from the left byline, striker Bishop met the ball at pace and Mattis fired home the rebound off a defender.
Bishop then saw a goal disallowed for offside as Bury looked to double their advantage, while at the other end Maheta Molango had a goal disallowed for the same offence.
Striker Glynn Hurst had a close-range strike blocked in the 33rd minute and midfielder Richie Baker blasted over the rebound when he should have scored.
Steve Evans was unlucky not to equalise on the stroke of half-time. His close-range shot was brilliantly saved by Manchester City loan keeper Kasper Schmeichel.
Bury were the better side in the second half and, had it not been for a string of smart stops from Ingham, the margin of victory could have been much greater.
Marc Pugh saw a free-kick curl just wide before Ingham saved twice from Hurst and then denied Bishop with his legs.
Wrexham had their chances to at least grab a share of the points but Matt Crowell saw a free-kick blocked and Evans saw a close-range header fly inches over the crossbar with seven minutes remaining.