After a positive opening by United, David De Gea allowed Shkodran Mustafi's header to slip through his hands, with Ander Herrera failing to clear in time on the line.
Arsenal's second goal saw Marcos Rojo try desperately to make up for his error in giving the ball to Alexandre Lacazette only to slide in and help the ball on its way into the back of the net.
United responded quickly both times, with Anthony Martial grabbing the first equaliser before a calamity at the back for the Gunners allowed Jesse Lingard to nip in and level.
Mourinho was forced into a number of changes by his ever-growing injury list, particularly in defence.
He selected Chris Smalling and Eric Bailly, despite neither being fully fit, while Rojo made his first appearance of the season and Diogo Dalot his first ever start in the Premier League.
Speaking after the game, Mourinho said: "Maybe we feel frustrated with the result, we feel frustrated with some mistakes, we feel frustrated with some moments where a little bit more quality would be needed but we have to be happy with the spirit, we have to be happy with the effort, the dynamic, the intensity. "The team had a big soul. We scored four goals and we drew 2-2. Even in matches like today where we play well, we always shoot ourselves.
"But good performances. I could speak about everyone. Especially in moments where we are not very happy with the result, we need happiness on the pitch."