The 41-year-old has been in interim charge since David Flitcroft left the Shakers after an 11-game winless run.
Brass, who spent 18 months as a player at Bury, was Flitcroft's assistant before he took up his former position.
Ryan Kidd will be assistant to Brass, while Chris Sedgwick will remain as first-team coach.
Chairman Stewart Day said: "There are several things that I believe we can improve on and off the field. This is why I felt that by going with Chris and Ryan we are able to get back on track to where we should be.
"I want us to win games, be successful, play attractive football and I also want performances from the team. Both Chris and Ryan know what we are looking for as the winning culture needs to be within the club."
Bury are 20th after Saturday's 4-2 loss against at Bristol Rovers, which was a club-record 11th league defeat in a row, and they are winless in 15 games in all competitions.
League One side Bury have moved Chris Brass from head of football operations to the role of head coach on a deal until the end of the season.