While Milan's vice president Adriano Galliani had conceded defeat in his chase of Silvestre earlier in the week, the deal was confirmed on Tuesday and the 28-year-old central defender, who played 20 games for Inter Milan last season, will move across town for the 2013/14 Serie A campaign.
Speaking after the deal was confirmed, Milan coach Massimiliano Allegri made it clear Silvestre had been a priority signing for him, with defenders Daniele Bonera, Philippe Mexes and Mattia De Sciglio all currently injured.
"We are generally not in a hurry to sign players," Allegri said.
"But Silvestre is a good player."
Milan's pre-season campaign will take a step up on Wednesday with Allegri's side set to take on Manchester City in the Audi Cup in Munich.
But Allegri has confirmed key striker Mario Balotelli will not be involved in Bavaria as Milan players who featured in the Confederations Cup will be given an extended rest prior to the new Serie A season.
"Unfortunately he (Balotelli) won't take part in this year's Audi Cup as he just came back from vacation as all the other internationals, who played the Confederations Cup," Allegri said.
"They have just been training for a couple of days so you won't see them on the pitch in these two games."
Milan are set for two games at Munich's Allianz Arena this week and will play either UEFA Champions League title-holders Bayern Munich or Brazilian club Sao Paulo on Thursday, depending on their result with Manchester City.
