Moyes said that his side merely did their best to stop Arsenal playing their possession football, which could have seen them dominate the game.
"I'm not here to make friends with other managers. My job is to win games for Everton," Moyes told Sky Sports News.
"I thought that [Wenger was disrespectful]. I thought we passed the ball around well. We weren't going to let Arsenal come and dominate the game. We were going to stop them from doing that.
"No one said you can't be competitive and I thought we played well. We restricted their passing and didn't let them into their flow as much."
The result left Arsenal in fourth and Everton in sixth in the Premier League.