No details of the length of contract agreed with the 32-year-old Serbian defender were revealed.
Vidic’s move will end an eight-year connection with United, with whom he has won five Premier League titles and one Champions League.
“Vidic is a world-class player. I’m extremely happy to have completed this deal to bring Nemanja to Milan,” Inter president Erick Thohir told the club’s official website.
“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.”
The former Spartak Moscow player’s existing deal at Old Trafford was due to expire at the end of this season and he had already announced early in February that he would not be signing a new contract.
“I want to challenge myself again and try to make the best of myself in the coming years,” Vidic explained at the time.