Spanish boss Unai Emery took over Arsenal in the summer, replacing Arsene Wenger who had been in charge for 22 years.
The Gunners finished sixth last season, missing out on a Champions League place, but they are well set for an improved finish this campaign.
Romelu Lukaku and Fred both hit the woodwork for the visitors, while goalkeeper Bernd Leno made saves to deny Lukaku, Paul Pogba and Marcus Rashford.
Emery told BBC Sport: “I am very proud of our players. They gave all they could against the best team at this moment.
“We were playing very competitive for 90 minutes and I think we deserved this victory. We can be happy today.
“We knew Manchester United could use different tactics but we wanted to impose our gameplan and I think we did that.
“Our goalkeeper is very important and sometimes the goalkeeper can save us, and today I’m very proud of his work.”