"Ashley Young is officially an Inter player," the club said in a statement.
The 34-year-old underwent a medical in Milan on Friday before completing a deal reported to be worth 1.5 million euros ($1.6 million) plus bonuses.
"Today has been a beautiful day, thank you to everyone for the welcome. It is an amazing feeling, it is like being part of a family," said Young, who will wear the number 15 shirt at Inter.
"My message to the fans is that we will win again, I have arrived here to do this."
Young spent the past eight-and-a-half seasons at Old Trafford since arriving from Aston Villa in 2011.
"You gave me the chance to play with legends, to win trophies, to work under the greatest manager in history and to be your captain," Young tweeted in a farewell message to United.
"Thank you for your support during the highs and the lows. I leave as one of you, see you again in an away end soon," he added in a note to United supporters.
Young, capped 39 times by England, made 261 appearances for United and scored 19 goals. He was club captain this season but only started 10 league games.
He won the 2013 Premier League title, the FA Cup in 2016 and both the League Cup and Europa League in 2017.
Young leaves United with the team in fifth, 27 points behind leaders Liverpool ahead of Sunday's meeting at Anfield.
Inter signed Romelu Lukaku from United last summer and also have Alexis Sanchez on loan from the club.
Antonio Conte's side are second in Serie A, two points behind champions Juventus.
Young becomes the third English player to play for Inter, following in the footsteps of striker Gerry Hitchens, who won the Scudetto in the 1960s, and midfielder Paul Ince, who played for two seasons from 1995 to 1997.