Bertolacci, considered one of the brightest talents of his generation, became the Rossoneri's first major signing of the summer when he joined the club on four-year deal worth a reported €20 million ($22.4m, £14.2m).
"A.C Milan communicate that Andrea Bertolacci has been officially signed from AS Roma. The midfielder has signed a contract until 30th June 2019," said a club statement.
In an earlier statement the 24-year-old Bertolacci - who has spent the past three seasons playing for Genoa, scoring six goals in 35 league appearances last term - said: "It is going to be a great feeling to arrive at the San Siro with a Milan shirt, I cannot wait to start.
"My first choice was Milan, there is an important project and I will be ready for training camp. I am 24 and I an still improve, I want to back up the good work I did last season and take Milan a long way."
Until last week Bertolacci had been co-owned by Genoa and Roma although the latter bought the player outright with a payment of €8.5 million to Genoa.
Roma's move preceded his sale to Milan, who in recent weeks have been trailing behind city rivals Inter in the summer transfer market after they captured France midfielder Geoffrey Kondogbia from Monaco.
The Nerazzurri remain linked to Atletico Madrid defender Joao Miranda and have recently welcomed Jeison Murillo from Granada having signed the Colombian international defender in January.
Milan are reportedly close to signing Sevilla striker Carlos Bacca as both Milan clubs look to rebuild after finishing in mid-table and missing out on Europe next season.
Rossoneri coach Filippo Inzaghi was fired just over a week ago and replaced by Sinisa Mihajlovic, who led Sampdoria to a seventh-place finish last season and a place in the Europa League.
Mihajlovic has a tough task ahead if he is to hoist Milan out of the doldrums.
But Bertolacci said he is willing to play in any position if it means pulling on the famous red and black shirt.
"I have spoken to Luca Antonelli, Alessio Cerci and the coach, and everyone wants to do well," he added.
"This club has an important history and I am happy to be here with lots of class players. Last season I played as a midfielder but I will do what the coach wants.
"This year will be important for all of us, I will try to do my best and hopefully we will finish high up the table."