Flitcroft was dismissed by the Championship strugglers, who he kept up the previous season, on November 30 following a 3-0 defeat to Birmingham City.
However, the 39-year-old has quickly found new employment with a side he made over 100 appearances for as a player.
Jepson will now leave the club, who sit 20th in England's fourth tier, with immediate effect.
In a statement on Bury's official website, chairman Stewart Day said: "I have been a long-time admirer of David and what he achieved at Barnsley was a credit to him and to his philosophy of hard work and dedication.
"I am confident that after meeting with David and his team that we have secured the best manager available to take us to the next level."
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