Vidic, 32, announced last month that he would leave Old Trafford when his current contract expired at the end of the season.
The Serbia defender - who will be in action against the Republic of Ireland on Sky Sports tonight - was linked with a number of major European clubs - among them Juventus, Monaco and Galatasaray, but has opted for a move to San Siro.
Inter president Erick Thohir told the club's 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.
"I would also like to take this opportunity to thank Manchester United, Nemanja and his entourage for the professional and co-operative approach they have shown throughout the negotiations."
Vidic, who signed for United in 2006 from Spartak Moscow, has won almost every major honour in the game during his time at Old Trafford including the Champions League in 2008, five Premier League titles and three League Cups.
Vidic's agent Silvano Martina says the Serb is 'excited' by the prospect of moving to Milan.
"Nemanja is excited about the choice he's made. He had already decided to leave Manchester United, and Inter was a really interesting option," said Martina.
"On January 1st negotiations began. That's when all the parties started talking. Nemanja loved everything about this project and hopes to carry the club forward.
"Nemanja is a professional though, so right now he wants to do well for Manchester United, honouring the club and the fans."
A statement issued by Vidic on the Manchester United website in February read: "It's the last year of my contract and I have had eight wonderful years here. My time at this great club will always rank as the best years of my career.
"I never could have imagined winning 15 trophies and I will certainly never forget that fantastic night in Moscow, memories that will live with me and the fans forever. However, 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.
"I am now going to focus all my efforts on playing for Manchester United and do the best I can for the team until the end of the season."