The Belgian, on loan from Manchester United, picked up the injury during Sunderland's 3-2 defeat by Crystal Palace on Saturday.
A scan on Tuesday confirmed the 21-year-old has suffered ankle ligament damage.
Januzaj has scored once for Sunderland in seven appearances and was sent off against Tottenham earlier this month.
Sunderland winger Adnan Januzaj has been ruled out for a minimum of six weeks with an ankle injury.