The young winger struck his first ever goals for the Red Devils to cancel out Craig Gardner's first-half effort and hand the visitors a much-needed three points
Sunderland had capitalized on some poor United defending early in the match to take the lead through Craig Gardner, but was made to rue missed opportunities as United's latest young star stole the show.
Januzaj opened his account in the 55th minute as he met a low cross from Patrice Evra and fired past Keiren Westwood from close range.
His second – which saw him become only the second United player after Ruud van Nistelrooy to score twice on his first Premier League start - came just after the hour mark with a crisp left-foot volley that nestled in the bottom corner.
Robin van Persie returned to league action having been on the bench for United's defeat to West Brom last weekend, Wayne Rooney started despite missing the team's midweek Champions League clash with Shakhtar Donetsk.
Caretaker manager Kevin Ball made two changes to the Sunderland side beaten by Liverpool, with Ondrej Celustka and Valentin Roberge coming in for Sebastian Larsson and Carlos Cuellar.
The home side made the perfect start with five minutes gone as Gardner took advantage of some poor defending from United to open the deadlock.
Emanuele Giaccherini fired a low cross from the right-hand side, Phil Jones and Nemanja Vidic both failed to clear the danger and Gardner pounced from seven yards out.
Januzaj and Nani were the biggest threat for the visitors and both spurned first-half chances.
Nani was first to send a long-range effort towards goal but his shot was wide, while Januzaj missed the target from the edge of the penalty area after neatly turning away from two defenders.
Sunderland had a fantastic opportunity to double its lead in the 34th minute but Giaccherini was denied by an excellent save from David De Gea. The Italian met a left-wing cross from Adam Johnson but the Spaniard made a save diving to his right.
Nani continued to be the main danger for United and he forced the first save out of Westwood in the 41st minute after meeting a Rafael da Silva cross. Sunderland failed to clear the cross and the Portugal international fired towards goal from a tight angle but the goalkeeper pushed it wide.
United was level 10 minutes into the second half through the impressive Januzaj, who fed the ball out to Evra on the left and was perfectly placed to convert the Frenchman's cross from inside the penalty area.
Van Persie tested Westwood with a low shot soon after but it was Januzaj who fired United ahead just after the hour.
Former United defender John O'Shea failed to clear a Nani cross from the right and the Belgian volleyed the loose ball across Westwood and into the bottom corner.
United should have extended its lead late on but Van Persie missed a clear sight of goal with a poor finish wide of Westwood's goal. It would matter little though, as the young Belgian did enough to ease the pressure on Moyes.