Former Juventus striker Mandzukic has agreed a deal until the of the season with the Rossoneri, though it includes the option for a one-year extension.
The 34-year-old arrives on a free transfer after terminating his contract with Al-Duhail, having joined the Qatari club from Juve last year.
He will offer support to fellow veteran attacker Zlatan Ibrahimovic, who has scored 12 goals in eight Serie A appearances this term.
Speaking about Mandzukic after scoring a double in a 2-0 win over Cagliari on Monday, Ibrahimovic told Sky Sport Italia: "I'm happy we have two that scare our opponents."
Mandzukic made 162 appearances in all competitions across four and a half years with Juve, scoring 44 goals and setting up another 19.