Craig Gardner had got the hosts off to the perfect start when Phil Jones and Nemanja Vidic both failed to clear their lines before the ball eventually fell to the midfielder who fired past David De Gea and into the bottom corner.
Kevin Ball's side almost doubled their lead midway through the first half when the lively Emanuele Giaccherini saw his header expertly palmed away by De Gea following Adam Johnson's cross from the left.
18-year-old Januzaj, who had impressed in only his third appearance for David Moyes' side, levelled proceedings when he guided Patrice Evra's cross past a helpless Keiren Westwood.
The Belgian youngster then fired the visitors into the lead on the hour mark when he volleyed across Westwood and into the bottom corner after Nani's cross was headed into his path by John O'Shea.
Robin van Persie had the chance to kill the game off in the closing minutes after Antonio Valencia played him through, but he poked wide with the outside of his boot despite being unmarked.
Sunderland's interim head coach Ball decided attack was the best form of defence and went for it from the off.
It took the hosts just five minutes to get their noses in front, as Giaccherini tricked his way past Evra on the Sunderland right all too easily and fired in a cross which Jones thumped straight at team-mate Vidic.
The Serbia international could do little more than present the ball to the fast-arriving Gardner, who drilled a low shot across keeper De Gea and into the bottom corner.
United's efforts to get back into the game were subdued at best. Januzaj fired in a series of shots from distance, most of which were blocked by a defence in which the recalled Valentin Roberge in particular proved impressively stubborn, while Wayne Rooney, back in the starting line-up after recovering from a shin injury, carelessly ran into an offside position as he attempted to get on the end of Evra's 14th-minute cross to the near post.
But Moyes' men were astonishingly fragile at the back, where Jones in particular was found wanting on more than one occasion, and the Black Cats could have gone in at the break with a significantly bigger lead to defend after it.
Gardner made the most of the defender's indecision to run in on goal once again with nine minutes gone, but with the unmarked Jozy Altidore pleading for a cross, he delayed and Jones managed to recover.
However, it was De Gea who came to the rescue 11 minutes before the break when he somehow managed to claw away Giaccherini's goal-bound header with a save of genuine quality.
Westwood was called upon to beat away Nani's 41st-minute shot, but Giaccherini might have doubled Sunderland's lead in the final minute of the first half when he was presented with another golden opportunity by Adam Johnson's run through a porous United defence, only to sky his effort high over the bar.
United's desperation was summed up within three minutes of the restart when Januzaj collapsed in a heap as he went past John O'Shea inside the penalty area and was promptly booked for diving.
But the Belgian was not to be denied and his industry paid off seven minutes later when he played the ball out to Evra on the left and then worked his way into the penalty area to meet the Frenchman's cross with an assured finish.
Westwood had to get down well to keep out a 58th-minute Van Persie snapshot, but he was powerless to prevent United taking the lead three minutes later.
O'Shea got his head to Nani's right-wing cross and the ball dropped nicely for Januzaj, who gleefully dispatched it into the bottom corner with a sweet left-foot volley.
Van Persie should have made sure at the death after being played in by substitute Valencia, put prodded his effort wide of the upright.