Andrew Tutte gave the Shakers an eighth-minute lead before Danny Nardiello converted a 50th-minute penalty and struck again three minutes later. Substitute Danny Rose added a fourth goal with two minutes remaining to put Flitcroft's men seven points behind seventh-placed York.
Bury have three games left to make up the deficit and visit the Minstermen on Easter Monday.
"I was quite emotional after the game," said Flitcroft. "It's exhilarating when you get a complete performance - and I think that was.
"I've asked a lot of the players and staff in recent months. We've put in the hard yards behind the scenes and days like today put the gloss on it.
"I was pleased to keep a clean sheet in the first half but I think the timing of the second goal was key. We were moving the ball around well and it demoralised them. They threw caution to the wind, they left themselves open, and we exploited that.
"On another day, Danny Rose could have had three. We finished with four and could have won it by more, so overall we dismantled a very good team."