The experienced Serbian defender will continue his career at the Giuseppe Meazza side after a successful eight-year spell at Old Trafford
The Serbian defender will join the Serie A outfit once his contract with the Premier Leage champions expires this summer, having agreed a two-year deal with the Nerazzurri, as revealed by Goal last month.
Monaco and Galatasaray also showed interest in signing Vidic, but the centre-back judged Inter's offer, which is believed to be worth €3 million (potentially rising to €3.5m) per year, to be the most appealing.
"Vidic is a world-class player. I'm extremely happy to have completed this deal to bring Nemanja to Milan," Nerazzurri president Erick Thohir stated.
"He's one of the world's best defenders and his qualities, international pedigree and charisma will be an asset to the team and help bring through our younger players. He will bring added value to the club and is another piece of the great Inter side we are building.
"I would also like to take this opportunity to thank Manchester United, Nemanja and his entourage for the professional and cooperative approach they have shown throughout the negotiations."
The 32-year-old Vidic started his professional career with Red Star Belgrade and also wore the jerseys of Spartak Subotica and Spartak Moscow before joining Manchester United in the summer of 2006.
He had since won five Premier League titles, three League Cups, the Champions League and the Club World Cup with the Red Devils.