Arnautovic was twice denied by home goalkeeper Loris Karius before half-time, while the Austrian also carried the ball into a number of dangerous positions as the Hammers took the game to the Reds in the opening 45 minutes.
Moyes felt his in-form No7 should have been more selfish and gone for goal himself, rather that looked for his teammates in the final third, only to see those attacks fizzle out.
“Marko has been really good, I've got to say, he's been great," Moyes told whufc.com. “We're demanding of him that his effort is big and he keeps his running stats up, which have been very good and physically, he does that job for the team.
“His ability is not in question and he is capable of doing anything on the ball. On Saturday, he got away a couple of times and I wanted him to go on his own, but he waited for support and laid a couple of balls off instead of going on his own against players I thought he had a chance to stand them up and take them on.
“He's doing a good job for us, he's a real threat, we like him a lot and we're really pleased with his work."