Kovac, 48, will be Monaco's sixth manager in just over 18 months after a long period of instability at the club.
"Our paths are separating earlier than expected but I would like to thank Robert Moreno for accepting the challenge," Monaco vice-president and general manager Oleg Petrov said in a statement on Sunday.
"With his staff, Robert did his utmost to improve the team, with enthusiasm and dedication. I wish him the best for the future."
Leonardo Jardim, Franck Passi, Thierry Henry, Jardim again and Moreno have all managed the club in the last 18 months.
Sources have told ESPN that players were unaware Moreno was going to be sacked when they trained with the former Spain head coach early on Saturday after beating Cercle Brugge 2-0 in a friendly on Friday.
Moreno, 42, was appointed last December to take over from Jardim after he was sacked from his second spell at the club.
With no experience as a club manager before, Moreno was at the helm for less than four months after the coronavirus pandemic halted Ligue 1 on March 13.
Kovac is due to arrive Monaco in the coming days and will have five weeks of preseason before the start of the 2020-21 Ligue 1 campaign.
After winning the league in 2017, Monaco has experienced a downturn of form finishing 17th and ninth in the last two seasons.
Paul Mitchell was appointed sporting director of the club in June with the mission of reducing the squad size and building a team capable of competing in the Champions League.
Replacing Moreno with Kovac is seen as Mitchell's first big move in achieving a return to competitive football.