Granit Xhaka fired Unai Emery's side in front when his shot swerved past a bamboozled David De Gea and into the back of the net, before Pierre-Emerick Aubameyang scored from the spot with 20 minutes remaining when Fred was deemed to have brought down Alexandre Lacazette inside the box.
In terms of the result, Solskjaer refused to blame De Gea for not keeping out Xhaka's curling effort, instead questioning why his defence afforded the Switzerland worldwide so much room.
"He's disappointed every time he concedes a goal because he's a top, top keeper", he said. "The ball moves so unusual and it looks like it goes to David's left and it changed direction", he said.
Solskjaer was initially handed the reins until the end of the season after replacing Jose Mourinho in December, but he has revived the club's form during his 18 games at the helm.
For Solskjaer, whose side enjoyed a stunning Champions League win at Paris Saint Germain in midweek, this was his first loss in 16 domestic matches since being appointed, ending a club-record run of nine successive away wins. It was important to press them as well.
"You don't want to see it again so we were motivated and we knew before the game about our chance because United had one more point than us".
United have had to contend with a host of injury problems during a packed recent fixture schedule, but Solskjaer would not accept that Sunday's result was down to a build-up of fatigue.
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"I thought we created some great chances, but we just didn't take them".
Making it even sweeter, Arsenal climbed above Manchester United into fourth place in the race to qualify for next season's Champions League.
And Spaniards David de Gea, Juan Mata and Ander Herrera got the chance to digest the disappointing result as they dined together at Cibo Restaurant in Hale.
We are, however, unbeaten in five in the league with only Tottenham taking points off us in the current run.
"To take three points and get a good position for our target is very good and I'm happy for the players".
At the moment, I think we are playing confidently in each match and today we showed everybody that.