Boro caretaker boss Steve Agnew brought Stewart Downing back into the team for his side's first game since the departure of Aitor Karanka, while United boss Jose Mourinho again started with a three-man defence.
Marcus Rashford was also handed a rare start for the Red Devils and had the first chance when he forced an early save from Victor Valdes.
The home side themselves had some nice moments, with Gaston Ramirez testing David De Gea but United's dominance finally paid off on the half hour when Marouane Fellaini headed home Ashley Young's deep cross.
United deservedly led as Boro continued to flounder in attack and doubled their lead on 62 minutes.
Jesse Lingard was the benefactor of some poor defending as, after picking the ball up around the halfway line, he carried the ball for some distance without being challenged before rifling home from 25 yards.
That seemed to suggest the points were safe for the Red Devils but Agnew made a couple of attacking changes by introducing Rudy Gestede and Adama Traore and his bravery paid off.
Gestede's introduction prompted his side to throw the ball into the box and he benefitted when Chris Smalling miscued a clearance into his path and the Benin international fired home.
Boro continued to push but conceded a third in stoppage time when former United man Valdes slipped as he was set to clear the ball and Antonio Valencia ran in to tap home.
The result plunges the Teessiders further into relegation trouble, while United keep themselves in the race for the top four.