A Jesse Lingard double and a strike from Antonio Valencia was enough for United who were put under pressure for large amounts of the game.
The Gunners had numerous chances but were met by an inspired David de Gea as Alexandre Lacazette scored the hosts only goal of the match.
There was a red card for Paul Pogba late in the second half, but that didn't stop United from earning all three points.
United ahead in double quick time
Arsenal came into the game perhaps as slight favourites, but for all of Man United's struggles in games against the top six they made a flying start at the Emirates, finding themselves 2-0 up inside 11 minutes.
It was defensive errors that were the Gunners downfall as two mistakes from the Arsenal back line were taken advantage of.
United opened the scoring when Laurent Koscielny's poor pass across field was intercepted by Valencia. The right wing back then exchanged passes with Pogba before advancing into the penalty area and firing underneath Petr Cech to make it 1-0.
For United's second, Shkodran Mustafi was caught in possession. The German centre half has been in excellent form of late and his error was hugely uncharacteristic of a so far brilliant campaign.
Lingard was able to steal the ball off of Mustafi, before playing a nice interchange with Anthony Martial and then finding the net.
Arsenal first half chances go begging
After conceding Arsenal were forced into making a change. Alex Iwobi came on for the injured Mustafi as Arsene Wenger's side reverted to four at the back. Somehow though Arsenal failed to score despite numerous chances.
Their first major opportunity came after 20 minutes. Mesut Ozil's corner wasn't dealt with and Granit Xhaka's header back across goal was almost turned home by Lacazette.
Ozil then had a chance of his own. He tried bending a free-kick over the United wall but Ashley Young ran back to head the ball away.
The tempo of the first half was frightening and Arsenal's attempts at goal didn't stop there. Lacazette saw a close range shot deflected onto the bar by De Gea, and then the follow up was fired wide by Xhaka.
Ozil then saw another effort denied with his volley from the edge of the area going straight at De Gea. Meanwhile Aaron Ramsey saw a shot blocked brilliantly by Nemanja Matic inside the area.
The Gunners were trying desperately to get themselves back into the game but De Gea was equal to everything they threw at him. Hector Bellerin's fierce drive was tipped away and then Sead Kolasinac's effort was saved once more.
The Spanish 'keeper saved his best stop until just before the break. A free-kick from the left was floated in towards the back post and the ball bounced off of Romelu Lukaku forcing De Gea into making a fantastic save. The follow up resulted in a goal mouth scramble but somehow the Gunners went into the break without scoring.
Gunners finally find the net
Arsenal came out for the second half in the same way they finished the first and it didn't take long before they finally found the goal.
Alexis Sanchez dinked a lovely ball into Ramsey in the middle who touched it into the path of Lacazette and he smashed the ball home without a United defender near him to make it 2-1 on 49 minutes.
The game wasn't showing any signs of slowing down though. Lingard went close for United, as Cech deflected his effort on to the post before the follow up was blocked well by Nacho Monreal on the line.
Arsenal went straight up the other end and almost scored an equaliser. Iwobi found space in the area before hitting a shot goal bound which was denied.
De Gea was in excellent form for Jose Mourinho's team and he was at his best to prevent Arsenal from scoring again. Lacaztte was able to turn his man and force a save and then somehow De Gea saved again from Sanchez's follow up.
Lingard kills the game off
Arsenal's attacking play in the game was outstanding, but their defensive display was very poor and again it was weak defending that led to another United goal as they added a third just after the hour mark.
The visitors were excellent on the counter all game and it was from another break way that they found the net. Lingard lead the attack before playing in Pogba down the right. He got the better of Koscielny all too easily and then slid the ball across goal for Lingard who was left with a tap in.
Pogba sees red but Arsenal denied penalty
As if the game hadn't produced enough moments Pogba received a red card with just under twenty minutes remaining.
The Frenchman left his studs in on Bellerin as they both went for the ball and referee Andre Marriner showed a straight red card for the challenge.
It was a moment in the game that prompted two very different substitutions.
Originally it looked as though Marcus Rashford was about to be brought on, but Mourinho opted for Matteo Darmian as he replaced the excellent Lingard. Meanwhile Olivier Giroud replaced Kolasinac for the hosts as Arsenal looked to throw everything at it.
The Gunners couldn't find their way back into the game however, but were denied a late opportunity to make things close.
Danny Welbeck was brought down inside the area with just a few minutes left but Marriner waved away protests as the Gunners fell to their first defeat at home this season.