Scot questions key decisions however, after contentious calls
The Toffees went 2-0 down after goals from Bacary Sagna and Cesc Fabregas before Tim Cahill pulled a goal back one minute from the end.
Although Everton caused Arsenal problems with their fightback, Moyes says it was too late to get back into the game.
"I didn’t think we played well enough to win the game. We did okay in the last 10 minutes or so, it's easy when you're 2-0 down to do that," he told Sky Sports.
"We conceded the goal just before half-time and the one after half-time made it really difficult.
"When you're not playing well you can't concede. We gave the ball away badly which allowed them the second goal.
"I think we had a couple of chances. [Jack] Rodwell, and Louis [Saha] had a bit of a chance. I knew we could get back in it if we got that goal five or 10 minutes earlier."
Arsenal's Sebastien Squillaci was fortunate not to receive his marching orders after a foul on Saha in the second half, and Moyes feels his side were unlucky with some key decisions in the match.
"I've not seen it again [Squillaci's foul] to pass comment. At the time I thought it's got to be close but it wasn't [a red card]," Moyes said.
"Most things didn’t quite go [our way]. People who've watched it and the people in the studio will probably feel that as well," he added.