Tom Soares headed the Shakers into a 33rd-minute lead, although the Dons had chances of their own.
The visitors continued to threaten after the break so a 77th-minute strike by substitute Ryan Lowe was welcome for fourth-placed Bury.
That kept the Shakers two points off the top three heading into next week’s final game.
Wimbledon made a lively start, with Sean Rigg twice denied by Bury goalkeeper Nick Pope.
Bury’s Natham Cameron then had a header saved before Chris Hussey’s corner found Soares at the near post and his glancing header beat James Shea.
The Dons replied with Rigg’s 20-yard strike forcing a fine low save from Pope.
Danny Nardiello then latched onto Pope’s punt forward to lob Shea, only to see it come back off a post.
Andrew Tutte should have doubled Bury’s lead 20 seconds after half-time as he blazed over and Adedeji Oshilaja almost equalised as he headed wide from George Francomb’s corner.
Rigg then volleyed wide before Lowe fired over, but the veteran striker made no mistake after Soares caught the Dons in possession, firing low past Shea to seal victory.