Ezri Konsa opened the scoring early before Gabriel doubled the hosts' advantage after the break.
Speaking to Everton TV, the tactician admitted he expects more from his players but believes aspects of their performance against Mikel Arteta's Gunners offered some encouragement.
“I thought there were better bits in the second half than in the first. We pressed them and got after it,” Moyes told club website.
“We unfortunately lost a goal when we were certainly back in the game at 1-0.
“There were lots of good things the players did today – defended well for long period but just had a couple of lapses.
“Obviously, the goal after five or six minutes wasn’t easy. And for the second one, I think a couple of people turned off for that action.
“I want more chances. Obviously, we’re playing against a really strong opposition. We might not get three, four, five, six chances, but we have to try to take them when we get them.”