United were looking to bounce back from successive home defeats by Tottenham Hotspur, in the league, and Swansea City, in the FA Cup, but instead they lost again, and to the Premier League's bottom side.
Ryan Giggs, who had earlier hit the crossbar, gifted Sunderland the lead with an own goal and despite Nemanja Vidic equalising early in the second half, Fabio Borini's 65th-minute penalty gave the hosts victory.
Having eliminated Chelsea in the previous round, Sunderland remain on course to reach a first domestic cup final since 1992, when they lost to Liverpool in the FA Cup final.
The importance of the fixture was evident in the fact that both managers made seven changes to the sides they fielded in the FA Cup at the weekend, although United were once again without the injured Wayne Rooney.
Eager to exploit any uncertainty in the visiting ranks, Sunderland looked to get balls into the box at the earliest opportunity and saw Borini drag an early shot wide from just outside the area.
Giggs sent a deflected 25-yard shot against the crossbar, while Januzaj had a goal ruled out for offside and Vidic planted a header wide from a corner shortly before half-time.
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