Former Bayern boss Kovac was fired last week with the Principality side sixth in the table, 17 points behind leaders Paris Saint-Germain. The 2017 Ligue 1 winners were knocked out of this season's Champions League qualifying stages with Kovac in charge.
Clement, 47, heads to Stade Louis II on a deal until June 2024 after two seasons with Belgian champions Club Brugge.
In a statement in English on the club website, chief executive Oleg Petrov said Clement's "proven ability to bring out the best in young and talented groups" made him the right person to continue the club's bid to bring Monaco "back into the elite of French football and onto the European stage in a sustainable manner".
According to Forbes magazine, Monaco's owner Dmitry Rybolovlev is worth $6.7 billion and the club have won just two trophies since he took over in 2011.