Golovin, who impressed for Russia at the 2018 World Cup, had been strongly linked with a move to Chelsea since the start of the month.
However, Monaco have announced that they have secured the midfielder's signature on a deal until the end of the 2022-23 campaign.
"We are very happy with the arrival of Aleksandr Golovin at AS Monaco. He is a talented young player, very prominent during the World Cup, but we have been following him for a long time," Monaco vice-president Vadim Vasilyev told the club's official website.
"He already has solid experience in Russian first division and internationally. Despite the competition of very large European clubs, Aleksandr has chosen the sporting project of AS Monaco, which will offer him the best conditions to continue his progress."
Meanwhile, Golovin added: "I am very happy to sign at AS Monaco. This is a new adventure for me that I approach with a lot of ambitions. I am now very much looking forward to meeting the staff and my new teammates and to get going."
Golovin scored seven times and registered six assists in 43 appearances for Monaco last term.