The 18-year-old attacking midfielder has signed a five-year deal in the principality but will spend the 2015-16 campaign in the Bundesliga.
Saint-Maximin, who made nine Ligue 1 appearances for Saint-Etienne last season, is the latest in a long line of highly talented young players to have joined Monaco in the close season.
They have already brought in the likes of Portuguese winger Ivan Cavaleiro from Benfica, the Malian Adama Traore from Lille, Fares Bahlouli from Lyon and signed Croatian midfielder Mario Pasalic on loan from Chelsea.
Those moves have helped strengthen a squad that has seen Geoffrey Kondogbia and Yannick Ferreira Carrasco depart for Inter Milan and Atletico Madrid respectively in big-money moves.
Monaco won 3-1 away to Swiss side Young Boys on Tuesday in the first leg of their Champions League third qualifying round tie.