Juventus had also reached two Champions League finals in the last three seasons, losing both to Barcelona and Real Madrid respectively.
Italy international Bonucci, capped 70 times, made 319 appearances for the Turin club.
He joined Milan last week for a fee of 42 million euros ($48.9 million) - the highest so far in this Serie A transfer window - subject to a medical.
As the club finalised the move on Thursday, Milan coach Vincenzo Montella described Bonucci as the best central defender in the world alongside Spain's Sergio Ramos.
"For me, it's a dream to be able to coach him," he said on Milan's website (www.acmilan.com).
Bonucci later said on Instagram that he wanted to "write a new chapter in the history of the club and in my career".
He added: "When you think of Milan, you immediately think of its history and all the champions who have worn the shirt.
"I think of (Franco) Baresi, (Alessandro) Costacurta, (Paolo) Maldini, (Alessandro) Nesta, (Mauro) Tassotti. I admire them for what they have given to football and this club."
Bonucci's decision to leave dominant Juventus for a club which has spent the last few seasons slumped in midtable mediocrity surprised many observers, who believed that he could have chosen a top English or Spanish side instead.
Milan, however, have gone on a spending spree following their sale in April to a Chinese-led consortium in a 740 million-euro deal.
They have also signed Portuguese striker Andre Silva, Turkish midfielder and free-kick specialist Hakan Calhanoglu, Swiss leftback Ricardo Rodriguez and Argentine midfielder Lucas Biglia.
Their total spending is already at 189.5 million euros according to the specialist website Transfermarkt, compared to 22 million euros in sales.
They have also signed 20-year-old Ivorian midfielder Franck Kessie, one of Africa's most promising players, on loan from fellow Serie A side Atalanta.
Milan finished sixth in Serie A last season, just high enough to qualify for the Europa League and end a three-season absence from European football.
($1 = 0.8595 euros)
(Reporting by Silvia Recchimuzzi in Gdynia and Brian Homewood in Zurich; editing by John Stonestreet)
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