Kovac, who has signed a three-year deal with the Ligue 1 club, was relieved of his duties by Bayern in November last year.
The Croatian led Bayern to a domestic double in his first season at Allianz Arena, but his position became untenable following a 5-1 defeat to his former side Eintracht Frankfurt.
Bayern have gone on to win another double this season under Hansi Flick and hold a comprehensive 3-0 advantage over Chelsea in the last 16 of the Champions League from the first leg.
Kovac replaces Robert Moreno at Monaco, with the club having confirmed the Spaniard's departure earlier on Sunday.
"We are very happy to welcome Niko to the club," said Monaco vice president Oleg Petrov.
"This season marks the start of a new chapter for Monaco, with an asserted sporting orientation in line with the club's objectives.
"This direction will be embodied by Niko as coach. Niko is recognised for his work both for the national team and for the club, in the development of young talent, the management of experienced players and the ability to achieve very good results. I'm convinced that his experience and personality will help us achieve our goals."
Monaco finished ninth in Ligue 1 in 2019-20, which was ended early due to the impact of coronavirus.