The 20-year-old Belgium winger was reportedly a target for several European clubs including Tottenham Hotspur, Olympique Marseille and Valencia.
"Adnan is a high quality and versatile player who complements our attack," Dortmund sports director Michael Zorc said.
Januzaj made four appearances for United this season, scoring in the 1-0 win over Aston Villa this month.
(Reporting by Karolos Grohmann, editing by Ed Osmond)
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