The Red Devils raced into a comfortable two-goal lead with a little over an hour on the clock, but Rudy Gestede pulled one back for the hosts to set up a tense finale.
United, who had dropped points in four of their last six games prior to today, managed to successfully hold on and were able to add a third at the death through an Antonio Valencia tap-in.
Marcus Rashford used his pace to get in behind Bernardo Espinoza with just five minutes played - a regular feature across the course of the afternoon - only for former United stopper Victor Valdes to pull off a big save.
Boro's struggles this term have largely been down to a lack of attacking threat, attempting a league-low 65 shots on target in the Premier League prior to today, and their only effort of the opening 45 minutes saw Gaston Ramirez - restored to the side by stand-in boss Steve Agnew - kept out by David de Gea from distance.
Valdes was proving to be in inspired form as the first half transpired, producing two more saves in quick succession to keep out Rashford from a few yards out as well as Valencia's follow-up attempt.
The deserved breakthrough did arrive for the visitors 30 minutes in, though, as Marouane Fellaini nodded home from an Ashley Young cross - despite four Boro players being inside the box to United's one - to end a run of 30 league games without scoring.
The Smoggies, still without a league win since prior to Christmas, did step things up slightly after falling behind but they were unable to create any real openings in the remainder of the first half.
It took until just after the hour mark for the game's next chance, which ended with Jesse Lingard blasting the ball into the top corner of the net when being allowed to advance to the edge of the Boro box.
Alvaro Negredo sent a header down the middle for De Gea to collect and glanced another narrowly wide, but the Smoggies were struggling to offer any serious response after conceding a killer second.
From pretty much nowhere they did have a route back into the game, however, with substitute Gestede prodding home from close range to end a run of 509 minutes without a goal for the home side.
The fourth and final goal of the afternoon went the way of United, though, as Valdes slipped when attempting to punt the ball upfield, allowing Valencia to roll into an empty on his 200th Premier League appearance to seal the win.
United move up to fifth in the table as a result of their victory, while Boro remain in the relegation mire and are now five points from safety with a game in hand to play.