The Hammers parted with a reported figure of £25m when they bought the Austrian from Stoke City in the January transfer window.
Since making the switch, the 28-year-old has made four appearances in all competitions for the East London outfit, and is yet to score or provide an assist in the Premier League.
Despite this, Bilic believes that the forward player can bring much more to the pitch, but he needs to stay unhappy "in a positive way".
"I could have asked for more, I want him to do more," Bilic told reporters. "He's got that quality and that something extra. I like him as a player but sometimes he gets happy with the few things that he does when I want him to push the borders.
"I know what he's capable of and I want him to keep molesting the opposition and never be happy in a positive way, but of course he did the job here."
Arnautovic has already seen red in his West Ham career, having been sent off in the team's 3-2 defeat to Southampton last month.