The 40-year-old was confirmed as the new boss of the Italian giants on Monday, succeeding former team-mate Clarence Seedorf who was relieved of his duties just five months after being appointed.
Inzaghi, who won two Serie A titles and two Champions Leagues in an 11 year-spell with the Rossoneri as a player, had been in charge of the club's Under-19 team for the last two seasons.
"I’m very happy," he told Milan Channel. "Today is an unforgettable day for me.
"I thank the president Berlusconi and the whole club for their confidence and the prestigious professional opportunity they have granted to me.
"I’m honoured and proud to be the coach of a team that has been my life for so many years. It is a team and a club with which I have shared joy and unforgettable moments.
"I will give everything for these colours, passion and determination, as I have always done both as a player and as coach of the youth teams."
"A hug to all Rossoneri fans. Forza Milan!"
Inzaghi has signed a two-year deal with the club.