Stitt was the head coach of Division II Colorado School of Mines for the past 15 seasons, going 108-62 with his innovative fly sweep offense. The Orediggers won three Rocky Mountain Athletic Conference titles under Stitt, who also won two RMAC Coach of the Year awards and regional coach of the year honors in 2004.
Colorado School of Mines finished 10-2 this season and ranked second in Division II with 378.3 passing yards per game and fourth with 522.2 total yards per game. Quarterback Justin Dvorak led Division II with 4,287 passing yards.
Stitt will replace Mick Delaney, who announced last month his intentions to retire at the end of the season after three years at the school.
Montana went 9-5 this season, finishing second in the Big Sky Conference. The Grizzlies lost 37-20 to Eastern Washington in the second round of the FCS playoffs on Dec. 6.
Montana has won two national titles in seven championship game appearances, plus 18 conference championships.