The Italy forward underwent a medical with the Grifoni on Thursday before completing his move to the Luigi Ferraris stadium.
A statement from Genoa read: "Genoa announces it has temporarily acquired the sporting rights of AC Milan striker Alessandro Matri."
Genoa needed to strengthen their attack after selling star striker Alberto Gilardino to Chinese side Guangzhou Evergrande.
"We knew that Gilardino could leave so we had completed a good move with regards to his replacement," Genoa president Enrico Preziosi said to Italian TV station Telenord.
The 29-year-old Matri spent the second half of last season on loan at Fiorentina and scored four goals in 15 league appearances for the Viola having struggled to cement a place in the Milan side.
Matri is eager to prove his worth at Genoa.
"I am honoured to be here," he said. "I want to start again and I am happy to be at the oldest club in Italy.
"I want to grow and be born again with this jersey.
"I chose Genoa for the shirt, for the history of this team and for its splendid fans.
"At the moment all I'm thinking about is doing well for Genoa, then maybe through my performances for Genoa I will get a call up from Italy."
Matri, who joined Milan last summer from Juventus, is under contract with the Rossoneri until June 2017.