Despite his contribution, Bilic demanded more from Arnautovic, who West Ham prised from Stoke City for a reported fee of 25 million pounds in the off season.
"I could have asked for more. I want him to do more. He's got that quality and that something extra," Bilic told British media after the game.
"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," the manager added.
Arnautovic was sent off in the 3-2 Premier League defeat to Southampton last month and Bilic said Tuesday's display marked the beginning of a comeback for the Austrian international.
"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," Bilic said.
West Ham, who drew 0-0 with West Bromwich Albion last week after posting their first Premier League win of the season against Huddersfield, host Tottenham Hotspur on Saturday.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by John O'Brien)
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