"My time with Milan finished in the best possible way, with a goal in the final game, so it was all perfect," Inzaghi, 38, told Italian media.
"Now I begin another dream, which is to be a coach and hopefully win the Champions League."
Inzaghi, who scored 156 Serie A goals in 370 games, added: "I couldn't go to play elsewhere, as I could not just leave Milan. Today I received a proposal from England, but I can't leave Milan.
"I signed a two-year contract and will coach the youth team, then I hope one day to lead the senior squad too."
Milan are undergoing a major rebuilding programme, with veterans Clarence Seedorf, Alessandro Nesta, Gennaro Gattuso and Mark van Bommel among those leaving the club.
However there was widespread frustration when star striker Zlatan Ibrahimovic and key defender Thiago Silva were sold to Paris Saint-Germain, although Milan insisted money would be ploughed into new signings and reducing the club's debt.