Cuisance has made 35 Bundesliga appearances over two seasons, tallying three assists.
"After two seasons with Borussia, Mickael Cuisance will leave to join Bayern Munich with immediate effect," Gladbach confirmed in a statement.
"The 20-year-old midfielder completed a medical in Munich on Friday, with both clubs reaching an agreement on the transfer deal on Saturday."
Cuisance has represented France at every youth level and started all four games for his country this summer at the U20 World Cup, where they fell 3-2 to the United States in the round of 16.
The midfielder will become the fifth French player on Bayern's roster, joining Kingsley Coman, Corentin Tolisso, Lucas Hernandez and Benjamin Pavard.
Like Cuisance, Hernandez and Pavard only joined the club this summer, with Ivan Perisic and Fiete Arp the club's other summer reinforcements thus far.
Cuisance will help the club strengthen in the middle of the pitch, joining Javi Martinez and Thiago Alcantara as players able to play as a holding midfielder.
"Mickael has his strengths in possession football. He has great technique, a strong left foot and a great mentality," Bayern sporting director Hasan Salihamidzic said of the midfielder.
Bayern failed to get their Bundesliga campaign off to a winning start on Friday, as they surprisingly drew Hertha 2-2 at the Allianz Arena.
Robert Lewandowski's brace was not enough to secure all three points for Niko Kovac's side, who nevertheless had a productive day off the pitch.
In addition to Cuisance's arrival, Bayern also confirmed the imminent signing of Philippe Coutinho from Barcelona on a one-year loan with a purchase option.
Bayern will look to earn their first Bundesliga win of the season next weekend when they travel to face Schalke.
Niko Kovac's side are looking to win their eighth consecutive Bundesliga title this season, with Borussia Dortmund the last club other than Bayern to lift the trophy when they accomplished the feat in 2011-12.