The 34-year-old had previously told United of his desire to join Inter and had not been involved in the club's past three games.
The deal brings an end to Young's eight-and-a-half-year stay at United.
He played 261 times for United and was club captain this season, but only started 10 league games in 2019-20.
Young said: “I am delighted to have joined a club with such a great tradition of competing and winning trophies at the very highest level and I'm looking forward to trying to help them continue that tradition.
“I had nine fantastic years at Manchester United and was truly honoured to have represented the club as their captain and I'd like to take this opportunity to thank the board and the fans for their support during that period.
“But this is the beginning of a new chapter for me and I'm really looking forward to the challenge."