The result means that both United and Arsenal stay in second and eighth place respectively.
I felt we had the bigger chances to win the game. Decent performances, good defensively, two massive chances in the second half," Solskjaer said after the game.
"We had a good performance. You come away with a share of possession, the bigger chances. There was a spell when we dropped off and were a little bit passive, but otherwise I'm pleased with the boys."
Arsenal made just one change to the side that beat Southampton during the week with Gabriel Martinelli coming in for Bukayo Saka, who Mikel Arteta said was left out as a precaution due to a sore hip. This meant the side was also without the in-form Kieran Tierney and top scorer Pierre-Emerick Aubameyang.
United made five changes after their midweek loss to Sheffield United.
The first half was scrappy with United enjoying the best opportunities. They had 67% of the possession and Arsenal struggled to threaten David de Gea's goal outside the first 15 minutes.
Marcus Rashford had the chance of the half when he received the ball in the six-yard box with just Bernd Leno to beat. However, he took one too many touches after Leno quickly closed him down and the opportunity came to nothing.
United continued to enjoy the bulk of the possession in the second half with Edinson Cavani given several opportunities to open the scoring.
Arsenal nearly upset the run of play just after the hour mark when Alexandre Lacazette hit the bar with a free kick and then Emile Smith-Rowe forced De Gea into a good save with a shot that was heading for the bottom right corner.
New Arsenal signing Martin Odegaard made his debut on 82 minutes, coming on for Smith-Rowe.
"I'm frustrated because we love to win. We tried our best, that's it. We have to try to improve every day, you have to be humble and to admit what you need to improve," Arsenal defender David Luiz said after the game.
"The boss is doing a great job, the young boys are fantastic and I think we are starting to create a real team with a real atmosphere.
"But if you don't work hard every single day things can change. The boys are happy because we have turned around a very difficult moment in our season."