A goal from Granit Xhaka and a second half penalty from Pierre-Emerick Aubameyang were enough to separate the sides in an entertaining fixture at the Emirates Stadium.
It was the Norwegian's first league defeat in 13 matches, a record which saw his side win 10 times and draw twice.
The loss is also Solskjaer's first away defeat in all competitions since replacing Jose Mourinho at Old Trafford in December 2018.
Solskjaer could only watch as his fatigued side wasted numerous chances to go ahead and then equalize the score.
Romelu Lukaku -- one of the heroes from midweek -- was particularly guilty of wasting golden opportunities in front of goal.
In addition, Arsenal stopper Bernd Leno was in inspired form and saved his side on more than one occasion. It was the first time Manchester United had failed to score in an away game this season.
At the other end, David de Gea should have done better with Xhaka's first half goal which seemed to confuse the wrong-footed Spaniard.
The win is Arsenal's ninth consectuive victory at home and it now sits just one point behind local rivals Tottenham in third.
"We just had to go for it towards the end, but we were not really at it and it is one of those days where you have to dust yourselves down from," said Solskjaer after the match.
There was another unfortunate flashpoint in the game when a fan ran onto the pitch to celebrate Arsenal's second goal. The spectator seemingly avoided pitch-side stewards before shoving Manchester United player Chris Smalling.