Arnautovic has scored five goals in his last six appearances, with the forward's resurgence coinciding with Moyes' appointment as Hammers boss in November.
And Moyes demanded more of the same from the Austrian, starting against West Brom at the London Stadium tonight.
"He was a big signing for the club and everybody expected big things from him as well," said Moyes.
"He wasn't a new recruit to the Premier League either, so you would have been hopeful that he could have hit the ground running.
"He's done really well for us, he has helped the team greatly. His energy and effort has rubbed off on the supporters as well so he's doing a good job for us.
"But it needs to be continued because we need him at the moment, he is the one who is really helping us so I hope his form continues."