The Belgian winger spent last season on loan at Sunderland and was allowed to leave Old Trafford by Jose Mourinho this summer.
Januzaj has signed a five-year contract for Sociedad and will wear the No. 8 shirt.
Posting on his Instagram he said: “Where to start… I’ve had some incredible moments with @manchesterunited since I joined.
“I want to say a massive thank you to all my teammates that have come and gone from the club.
“It’s been a pleasure to play with you all. Thank you to some of the coaches and staff that work at Manchester United to make it such a great club!
“Unfortunately, my time at the club has come to an end and it didn’t go the way I wanted it to go but I will always have love for Manchester United.
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“And finally, a huge thank you to the fans that have stuck by me through the good and bad times! I will never forget that positive support. Thank you #GGMU.”
Januzaj broke into United’s first-team when David Moyes took over from Sir Alex Ferguson at Old Trafford.
But he was unable to replicate his form of that season under Louis van Gaal or Jose Mourinho.
United have inserted a buy-back clause into the transfer agreement.
