Arsenal dominated for the majority of the game but fell behind in the second half thanks to an unfortunate own goal from Kieran Gibbs.
Wayne Rooney seemed to have put the game beyond the hosts when he capped off a swift breakaway to score late on.
Substitute Olivier Giroud set up a tense few moments when he pulled a goal back in stoppage time but United held out to claim a second successive Premier League win.
Louis van Gaal admitted prior to kick-off he had been forced into playing a back-three, with the inexperienced Paddy McNair joining Tyler Blackett and Chris Smalling, and United were fortunate to keep the score goalless at half-time as the trio faltered.
Left wing-back Luke Shaw was also forced off with a knock in the opening 30 minutes, as Arsenal rained shots at David de Gea, and the Spaniard turned out to be the away side’s man-of-the-match.
Jack Wilshere, who was also forced off later with a possible ankle injury, missed a glorious chance from 10 yards that De Gea palmed away and Danny Welbeck, playing against his former club for the first time, also had chances to beat him
But left-back Gibbs collided with goalkeeper Wojcech Szczesny in the build up to the opener, allowing Antonio Valencia to drill a shot that found the far corner off Gibbs’ outstretched leg on the floor.
Arsenal poured bodies forward in search of an equaliser and the inevitable space left at the back allowed Angel di Maria to break quickly and set up Rooney to chip Damian Martinez, who had replaced the injured Szczesny, from close-range.
Giroud came off the bench to make a long-awaited return from injury and his superb strike into the top corner set up a tense period in eight minutes of added time.
But United moved into the top four with a fifth Premier League victory in nine games.