The 25-year-old defender has signed a one-and-a-half year contract after passing a medical and agreeing personal terms.
Haas has made 62 appearances for Bayern's II team, who play in the third tier of German football.
Mowbray said: "Maximilian was very keen to come here and test himself in English football. It's a challenge for him and his arrival will create additional competition for places in that area of the pitch.
"He is a real competitor and comfortable with the ball. The challenge for him is acclimatise to the English game and then take any opportunity that comes his way."
Haas' arrival follows the departures of David Wheater to Bolton and Gary O'Neil to West Ham.