French media reported Monaco paid around 30 million euros (26.68 million pounds) for the 22-year-old, who played an influential role as Russia reached the World Cup quarter-finals on home soil.
The versatile midfielder contributed a goal and two assists as the home team beat Saudi Arabia, Egypt and Spain before losing to Croatia on penalties in the last eight.
"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 and chief executive Vadim Vasilyev said in a statement.
"He already has solid experience in the 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."
In his three seasons with CSKA, Golovin played 113 matches, including 23 appearances in the Champions League and Europa League. He won the Russian league title once in 2016.
Golovin was linked by British media with both Chelsea and Arsenal in the close season window but the player chose the French club to begin a new chapter in a promising career.
"This is a new adventure for me that I approach with a lot of ambitions," Golovin said. "I am now very much looking forward to meeting the staff and my new team mates and to get going."
Golovin will boost Monaco's midfield options after they sold Joao Moutinho and Thomas Lemar to Wolverhampton Wanderers and Atletico Madrid, respectively.
(Additional reporting by Hardik Vyas in Bengaluru; Editing by Toby Davis)
