On a night of few clear chances for either team a second half penalty from Pierre-Emerick Aubameyang secured a crucial three points for the Gunners.
Neither side created much in the early stages on a rainy night in Manchester before a chance apiece just before the break.
United striker Mason Greenwood wriggled away from Gabriel inside the box but Bernd Leno reacted smartly to block his effort.
Arsenal’s best chance of the first 45 minutes fell to Bukayo Saka but the teenager could only head over from Hector Bellerin’s cross.
The visitors picked up the initiative after the break with Alexandre Lacazette releasing Aubameyang to release Aubameyang who curled just wide.
With neither really able to take charge of the game, the match turned on a decisive moment on 65 minutes.
Bellerin collided with Paul Pogba inside the box and despite the Frenchman’s protests, referee Mike Dean pointed straight to the spot and Aubameyang confidently slotted past David De Gea.
United almost pinched a point in the closing stages as Donny van de Beek’s effort deflected onto the post but Ole Gunnar Solskjaer’s side were unable to stage a big finale.
Full Time at Old Trafford it’s Manchester United 0 Arsenal 1