Goals from Granit Xhaka and Pierre-Emerick Aubameyang earned the Gunners a victory that sent them into the Champions League places while pushing United back down to fifth.
It was United's first league defeat since Dec. 16, Jose Mourinho's last game in charge, and ended Solskjaer's remarkable unbeaten start in the league as United's manager.
Arsenal went ahead in the 13th minute when Xhaka belted a swerving shot from 30 metres out, with United keeper David de Gea seemingly wrong-footed by the curve of the ball.
United hit the woodwork either side of the goal, with Romelu Lukaku heading against the crossbar first and Fred striking the post minutes later.
Lukaku had a golden opportunity to score from close range early in the second half but his first touch was heavy and, as he lunged to shoot, goalkeeper Bernd Leno blocked the shot with his body to send the ball spinning up and over the bar.
Aubameyang then converted the penalty, making up for his missed spot kick at Tottenham Hotspur a week earlier.
The win lifted Arsenal to 60 points, just one point behind their north London rivals Tottenham Hotspur, who are third, and two points above United who slipped to fifth.
Earlier on Sunday, second-placed Liverpool had beaten Burnley 4-2 to move on to 73 points, one behind leaders Manchester City.