Golovin, 22, had been linked with a move to Chelsea, Juventus and Barcelona after helping Russia reach the last eight at the World Cup finals.
But Monaco have swooped to secure Golovin as a replacement for Joao Moutinho, who joined Wolves earlier this week.
"I am very happy to sign at AS Monaco," said Golovin, who has signed a five-year deal with the former French champions.
"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."
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 at the World CUp.
Monaco vice-president Vadim Vasilyev said: "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.
"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."