Flitcroft switched to an adventurous 3-4-1-2 formation, which resulted in Anton Forrester and Nathan Cameron scoring inside the opening 19 minutes.
And although York substitute Michael Coulson pulled one back, the hosts hung on to give Flitcroft his first win as Bury boss and climb out of the League Two drop zone.
"You come up against a lot of rigid 4-4-2 teams but we got between them and caused problems," Flitcroft said.
"We worked on it, it's not lucky, and that's what's buzzed me up here. The lads have been really responsive to what I've asked them to do. I was pleased about many aspects of our play.
"The first goal was everything I believe, everything I prophesise and everything I want us to do on the pitch.
"And Ashley Grimes' pass for Anton Forrester's chance, Danny Nardiello's to Shaun Harrad - they're the things you go to bed and dream about.
"I was delighted with the way the game plan turned out as we got the goals from it. Danny Mayor was key to that from an offensive situation (playing behind the front two) but his defensive responsibilities were excellent too."