Initially joining the Diavolo on a co-ownership deal from parent club Genoa, he was signed outright this summer.
"The Rossoneri netted him thanks to some speedy business," representative Roberto La Florio told Sky Sport Italia.
"He was being watched by Inter and Juventus, but Milan Vice-President Adriano Galliani, with a lightening strike, signed him."
The 19-year-old only scored two Serie A goals last term, he already has four in seven games this time around.
"It was clear that Milan would focus on him after they sold Zlatan Ibrahimovic," La Florio continued. "He's certainly very happy with how things are going."