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 trophy.
"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."
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.
"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."
It will be Vidic's first spell in Italy's Serie A and the Serbian joins a Nerazzurri side who are rebuilding following Thohir's takeover of the club last November.
Inter were the last Italian side to win the Champions League, under former coach Jose Mourinho in 2010, but a disastrous ninth-place finish last season under Andrea Stramaccioni saw them miss out on Europe this season.
Ahead of Sunday's home game against Torino, Inter sit fifth in the table, 11 points behind the last Champions League qualifying place occupied by third-placed Napoli and 28 behind leaders Juventus.
Vidic arrived at Old Trafford in 2006 following a £7 million ($11.4m, 8.4m euros) move Spartak Moscow.
He became a key figure in nearly all of United's subsequent trophy triumphs and last term captained United to an historic 20th English title -- his fifth and the club's last under Alex Ferguson before the celebrated Scottish manager retired.
However, he made only 19 league appearances last season after he was laid low by a severe knee injury first suffered in a shock Champions League defeat by Swiss side Basel in 2011 that led to two major operations.
In the view of some pundits, Vidic has never been the same player since nor has his once dominant central defensive partnership with England's Rio Ferdinand, who could also be leaving Old Trafford at the end of the season, been as commanding.
However Thohir, whose first major signing as club owner was Lazio midfielder Hernanes, during the January transfer window, believes Vidic will prove to be a good addition to the side.
"He will bring added value to the club and is another piece of the great Inter side we are building," said the Indonesian businessman.
United have struggled under new manager David Moyes.
They have been knocked out of both domestic cups and are currently 18 points behind Premier League leaders Chelsea.
COMMENTS
Comments are moderated and generally will be posted if they are on-topic and not abusive.
For more information, please see our Comments FAQ