Arnautovic, who returned to the Hammers lineup after injury, was on the scoresheet alongside Javier Hernandez as Moyes's team moved into 12th in the Premier League table, level on points with the Hornets.
"You can see the difference Marko makes to us going forward," Moyes told reporters. "He's got power, pace and quality on the ball and he's doing a really good job.
"He's getting told he has got to keep at it and he's getting pushed all the way. We're telling him he is doing well for us, but he's not getting off the hook. I think he's got a bit to go.
"I told him he should have scored more goals, but he should also have had a penalty kick for a foul on him in the first half."
West Ham's next fixture is against Liverpool in the league later this month.