Inter have not released the details of the deal but the defender is believed to have put pen to paper on a three-year deal with the San Siro side.
Inter president Erick Thohir told the official Inter Milan website: "Vidic is a world-class player. I'm extremely happy to have completed this deal to bring Nemanja to Milan.
"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."
It was confirmed on February 7 that Vidic would leave United at the end of the season.
The captain said at the time, in a statement released by United: ''I have decided that I will move on at the end of this season.
"I want to challenge myself again and try to make the best of myself in the coming years.
"I'm not considering staying in England as the only club I ever wanted to play for here is Manchester United and I was lucky enough to be part of this club for so many years.
"I've got a few options to move on and I will choose the right one for me and for my family."
The player has been one of the most popular members of the Red Devils' squad since he arrived from Spartak Moscow for £7 million - a transfer bizarrely announced on Christmas Day in 2005.
He has won five Premier League titles, three League Cups and one Champions League title with United.