Destro, who was granted leave by Roma a day prior, underwent a medical at La Madonnina in Milan on Thursday, ahead of an initial loan deal until the end of the season with the option of a permanent transfer.
The 23-year-old Italy international is relishing the opportunity to lead the 18-time Italian champions back to the top, starting against Parma on Sunday.
Milan sit 11th in the Serie A standings, 23 points adrift of league leaders and reigning champions Juventus.
"It was a wonderful welcome. I am very happy to be here and to be a part of this group," Destro told Milan's official website.
"I have already spoken to coach [Fillipo] Inzaghi, there is a big desire to do well and to show that Milan are a great side, starting as of this Sunday.
"I cannot wait to get out on the pitch and play. The coach will decide my role.
"This is a big occasion for me, I cannot wait to play and get to know everyone. I want to say a big hello to the fans, we will do everything to do really well in the league."
Destro has struggled for regular game time for Roma this season, with the forward making 16 league appearances - half of those from the bench.
In that time, Destro has scored five goals for Rudi Garcia's championship hopefuls.
