Minamino scored 11 goals and provided two assists for Liverpool across all competitions last season
The Ligue 1 side were one of a number of clubs interested in the 27-year-old, who joined the Reds from Red Bull Salzburg for £7.25m in 2020.
Minamino said: "I am very happy to be part of this very exciting project."
He scored 14 times in 55 outings for Liverpool and was their top scorer in domestic cup competitions last season.
Minamino made just 11 appearances, including one start, in the Premier League last season and the deal for the forward is worth a fixed 15m euros (£12.9m) and 3m euros (£2.6m) in add-ons.
"It is a great joy for me to join AS Monaco, a club with a great tradition and among the most recognised in Ligue 1," Minamino said. "I'm looking forward to discovering my new environment and doing everything I can to help the team."
Liverpool boss Jurgen Klopp said: "It's hard to see Taki go, but it's a great move for him and one he thoroughly deserves. [He is] An amazing professional; super, talented player.
"As a person, he is full of warmth and makes everyone feel positive. A manager's dream, to be honest."
Minamino's departure came after Sadio Mane completed a £35m move to Bayern Munich and following the arrival on Merseyside of Uruguay striker Darwin Nunez.
-- Courtesy of BBC Sport