The Belgium winger, thought to have been surplus to Jose Mourinho's requirements at Old Trafford, was not included in the Red Devils squad for their pre-season tour of the United States.
The 22-year-old linked up once again with Moyes at Sunderland last season, joining on a season-long loan, but did not manage to score in 25 appearances as they were relegated from the Premier League.
A statement on the United website read: "Adnan Januzaj has completed a transfer to Real Sociedad. Everybody at Manchester United wishes Adnan well for the next chapter of his career."
Sociedad finished sixth in La Liga last season and will compete in the Europa League group stage in 2017-18.