The Italian club had been widely touted as the likely destination for the Serbia centre-back since it was revealed last month that he would leave Old Trafford at the end of the campaign.
Inter confirmed on Wednesday that they had got their man, with club president Erick Thohir immediately hailing the signing as "crucial" for the team.
"Vidic is a great champion," he said on inter.it. "I am extremely satisfied with the completion of this agreement that will bring Nemanja to Milan.
"He is one of the strongest defenders in the world, for his characteristics, his international experience, his charisma as a leader will be crucial for the team and the growth of our younger players.
"He will add value to the club and will be another pillar for the construction of a great Inter.
"Finally, I want to thank Manchester United, Nemanja and his entourage for his availability and the professionalism shown during the negotiation process."
Vidic, who joined the Red Devils from Spartak Moscow in January 2006, confirmed last month that he would not remain in England after the summer.
''I want to challenge myself again and try to make the best of myself in the coming years," he said in a statement released by United.
"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.''
Last season, Vidic skippered United to their historic 20th championship - his fifth and the last under former manager Sir Alex Ferguson.
Vidic has also won one Champions League trophy, three League Cups and a FIFA Club World Cup crown during his eight-year spell with United, and has started in 18 Barclays Premier League games and four Champions League matches this term