The 27-year-old arrived at Anfield from Red Bull Salzburg in January 2020 but struggled for regular starts in Jurgen Klopp's star-studded frontline.
Mohamed Salah and Sadio Mane were both ahead of Minamino in the pecking order, and he was pushed further down following the arrivals of Diogo Jota and Luis Diaz.
Minamino only had two years left to run on his Liverpool deal and was thought to be on the radar of several clubs, including Leeds United, Lazio and Lyon.
However, Monaco quickly moved to the head of the queue for Minamino's signature, and the Ligue 1 outfit have now wrapped up a £15.5m deal for the Japan international.
Klopp paid tribute to the "amazing professional" following confirmation of his departure, telling liverpoolfc.com: "It's hard to see Taki go, but it's a great move for him and one he thoroughly deserves.
"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. I'm sure there will be those who think it didn't quite work out. Those who think this are wrong. I don't accept that.
"His contribution far outweighs the opportunities we were able to give, in terms of starting matches. He made us better each and every day he was with us - not just in the games he played in but every single session in training. Perfect attitude, winner's mentality.
"His performances and his goals are the reason we have had to redecorate the Champions Wall so often. His achievements here will stand the test of time.
"I know he will be a big success at Monaco. I'm sure we will look on with a mix of celebration and envy as he progresses and flourishes there. He leaves with our appreciation and best wishes. Thank you, Taki."
Monaco boss Philippe Clement had confirmed the imminent arrival of Minamino over the weekend, telling AFP: "I'm very happy, I've known him for a long time. He was very high on our list. Everyone here is convinced that he has the qualities to bring something to the team.
"He already has experience and has done a lot of good things at Salzburg, in a style close to ours, just like at Liverpool where there is a lot of competition. That's why this is an opportunity."
Minamino - who has signed a four-year deal at Monaco - leaves Liverpool after scoring 14 goals and setting up three more in 55 appearances for the club in all competitions.
The former Red Bull Salzburg starlet ended the 2021-22 season with seven goals across the FA Cup and EFL Cup - lifting both trophies - while he also won the Premier League in the 2019-20 season.
Minamino made 11 Premier League appearances last term but only started one of them, which saw him find the back of the net in a 2-1 win versus former loan club Southampton.
The Japanese attacker follows Mane, Divock Origi and Loris Karius out of the door this summer, with Origi now closing in on a switch to AC Milan on a free transfer.