Aubameyang ended a run of five games without a goal by sending David de Gea the wrong way on 69 minutes after Paul Pogba had fouled Hector Bellerin.
United tried to rally but came no closer to an equaliser than when Donny van de Beek's cross-shot flicked off Mohamed Elneny and then into the face of goalkeeper Bernd Leno before bouncing to safety off a post.
It was not the greatest contest between two clubs who used to fight it out for the biggest prizes but are now a pale imitation of their former selves.
While Arsenal will be delighted at the deserved victory, this was not the kind of United performance that did justice to the memory of club legend Nobby Stiles, who died on Friday.
Pogba is the most high-profile figure in the United midfield that World Cup winner Stiles used to operate in. But the Frenchman had a day to forget, giving the ball away too cheaply when his side attacked and needlessly fouling Bellerin to turn the contest Arsenal's way.
The Gunners now have 12 points and are up to eighth and United remain in 15th on seven points.
It is only two weeks since Arsenal visited Manchester and did quite well, only to lose to a Raheem Sterling goal at City. Last week, United and another London club, Chelsea, played out a thoroughly forgettable goalless draw at Old Trafford.
The first-half of this contest felt like a combination of the two games.
United were a shadow of the side that hammered RB Leipzig in the Champions League on Tuesday. Their one decent opportunity came when Marcus Rashford found Mason Greenwood with a superb pass across the area but Bernd Leno saved with his feet and would have been disappointed if he had been beaten at the near post.
Arsenal were decent. Egypt midfielder Elneny justified his selection ahead of Granit Xhaka with an excellent performance screening the visitors' defence. William schemed ahead of him and Arsenal enjoyed the majority of possession.
But they never really looked as though they believed they could score.
Alexandre Lacazette might have done had Victor Lindelof not nudged Aubameyang's pass away from him.
Willian skimmed the bar with shortly afterwards and Bukayo Saka should have done far better when Bellerin picked him out with a right-wing cross.
Aubameyang also lacked conviction when he curled an excellent chance wide after the break but given an opportunity from the spot, the former Borussia Dortmund man delivered, in the process ending Arsenal's unwanted 29-game wait for an away win against a fellow 'big six' rival.
Solskjaer's not celebrating his century
This was Ole Gunnar Solskjaer's 100th game in charge of United - and it still feels like they are taking a step forward, before another going backwards immediately follows.
Over the past two weeks, United have been excellent at Newcastle, superb in Paris and brilliant against RB Leipzig. In between they were tepid against Chelsea and worse in this game. Add in the 6-1 home defeat by Tottenham just before the international break and it is little wonder questions remain about Solskjaer being the right man for the job.
He stuck with the same diamond formation that worked perfectly against Leipzig, only to find it completely ineffective against Mikel Arteta's well-drilled side, who did not allow United the space they had enjoyed in midweek.
In addition to Pogba, Bruno Fernandes was anonymous and it can only be concluded Edinson Cavani is still not properly fit, otherwise he would have been introduced far earlier than he was.
As it is, United are only three places above the relegation zone and while the season is still early - and they do have a match in hand - they remain without a home win.
It is the first time this has happened since 1972 and the gap to the clubs for whom they are expected to challenge for honours with is growing.
'We didn't turn up' - reaction
Ole Gunnar Solskjaer speaking to BBC's Match of the Day: "That's football for you. It goes up and down.
"We didn't turn up, especially in the first half. That's what we have to address because we know these boys can be so good.
"We knew that this was the big game this week. And unfortunately in the first half we didn't play well enough.
"Second half we played better, we never seemed to have to defend too much. But when we did have to, we were too slow and gave away the penalty."
"The boy is running away from goal, there is no need to go for the ball. Not much of a touch, but it's probably a penalty.
"It's for other people to judge if we're making progress."
United midfielder Paul Pogba: "We know it's a poor performance. Myself, I cannot give a foul away like this.
"I thought I would touch the ball, but I didn't. (I) cost us the goal today with the penalty.
"Like I say, details. We have to do better, the team, myself, it starts with me.
"I felt like I touched him a bit, I knew I was in the box, I shouldn't have given a penalty away like that.
"Maybe I was a bit out of breath, it made me do this stupid mistake."
Arsenal boss Mikel Arteta speaking to Match of the Day: "I wanted to generate belief and I asked them to be who we are as a team right now.
"I'm delighted because we played with a lot of courage and big personality.
"They played a really tricky formation with four midfielders and two that constantly strain your back four. We could have scored twice in the first half and when we scored the goal we had to manage those moments. There was high stress, it is different to any other ground.
"Individually you have to win the fights and step up to the occasion and the performance of every player today was top drawer."
United's home slump - the stats
- Only Chelsea (18) have won more Premier League games against United than Arsenal, with this their 16th victory against the Red Devils.
- United have taken just seven points from their six Premier League games this season - their lowest tally at this stage since David Moyes' short stint at the helm in 2013-14 (seven), a campaign in which they went on to finish seventh.
- United have failed to win any of their opening four top-flight games at Old Trafford for the first time since 1972-73 - a campaign in which they finished 18th.
- United have conceded five penalties in their past seven Premier League home games; the previous five penalties they had conceded at Old Trafford in the competition came over a period of 101 games.
- Pogba has conceded three penalties in the Premier League under Solskjaer, while no other Manchester United player has conceded more than one.
- Aubameyang is the first Arsenal player to score a penalty at Old Trafford in the Premier League - neither of the previous two attempts were converted (Robin Van Persie in 2011 and Gilberto Silva in 2006).
What's next?
United are next in action in the Champions League, when they have a trip to Istanbul Basaksehir on Wednesday (17:55 GMT). They visit Everton in the Premier League on 7 November (12:30 GMT). Arsenal play Solskjaer's former club Molde in the Europa League on Thursday (20:00 GMT) and then Aston Villa at the Emirates in the Premier League on 8 November (19:15 GMT).