Tom Soares headed the Shakers into a 33rd-minute header, 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 day.
Flitcroft said: “Wimbledon set up in a very unorthodox shape and for the first 20 minutes we were a bit disjointed.
“Nick Pope made two brilliant saves to keep a clean sheet, but after we scored – certainly after the second goal – I felt we controlled the game.
“We had several chances to win the match and we had to go in for the kill At 2-0, I think Wimbledon felt they’d had their best effort in that first 20 minutes.
“It looked apprehensive but I think that was Wimbledon’s buoyancy, coming here with nothing to lose. Other teams have come here with a fear factor and parked the bus but Wimbledon didn’t do that.
“I really wanted to take it down to that last day, it’s something we all look forward to. Next week we have a fantastic opportunity to take care of our own business.
“We did that so it’s not over yet and we’ve got one game to go where we’ve got to be ultra-professional.”
Wimbledon made a lively start, with Sean Rigg twice denied by Bury keeper Pope.
Chris Hussey’s inswinging corner then found Soares at the near post and his glancing header beat James Shea for the opener.
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 a corner.
Lowe then fired over but the veteran striker made no mistake after Soares caught the Dons in possession, firing low past Shea to seal victory.
Dons boss Neal Ardley said: “First half we were excellent. Second half was a bit more bitty. An individual error cost us the second goal and I think we packed in at that point.
“There wasn’t enough spine and character or fight and grit. That’s why Bury are up there and we’re mid-table. They know how to win games when they play well, we don’t.
“I was angry at half-time because it was the same old story. The last five or six games have been that way. We’ve been the better team in most of them but there are too many vital ingredients missing from the team.
“We haven’t got enough qualities to take the lead in games or see games out. The responsibility for that is down to me so I have to take the blame.
“I haven’t got those qualities in the team so I have to find those qualities.”