The Russia international had been close to a move to Chelsea and seemed to be on the way to Stamford Bridge as recently as last week, as former team-mate Sergey Chepchugov wished him good luck in England
But Monaco had already entered the race by making an offer and have now wrapped up the deal, with Golovin signing a five-year contract with the Ligue 1 side.
The deal will cost Monaco €30m up front, Goal understands, but CSKA will receive a percentage of any future sale of the 22-year-old star.
"I am very happy to sign for AS Monaco," he told his new team's website. "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 played 113 games for CSKA since making his senior debut in 2014 and won the Premier League title in 2016.
An Under-17 Euros winner in 2013, he has gone on to make 23 appearances for Russia's senior side. Golovin scored once and got two assists as his side reached the quarter-finals of the World Cup before losing to Croatia in a penalty shoot-out. His impressive displays on the global stage gave his reputation a boost, drawing links to Chelsea and Juventus.
"We are very happy with the arrival of Aleksandr Golovin at AS Monaco," vice-president Vadim Vasilyev said. "He is a talented young player, very prominent during the World Cup, but we have been following him for a long time.
"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."
Golovin joins defender Pele, midfielder Jean-Eudes Aholou and teenager Willem Geubbels among Monaco's summer signings, although Thomas Lemar has been sold to Atletico Madrid and Fabinho joined Liverpool.