West Ham were without their £25 million club-record signing for three games after the Austria international picked up a straight red card for an elbow on Jack Stephens.
It was only the 28-year-old's second game for the Irons since joining from Stoke City in the summer but Arnautovic is back in contention for Saturday's trip to West Bromwich Albion.
And although Bilic concedes that his big-money acquisition "did something very bad", the Hammers boss has called on him to make up for it by way of goals and assists.
"It's not a one-man game but he did something very bad at the start of his West Ham career," he told The Belfast Telegraph.
"Instead of helping us straight away he wasn't there for three games because of what he did against Southampton. So now he has to make it up to us.
"I've spoken to him a few times about that and he's said he can't wait to come back and that he wants to make it up to us. I'm expecting that from him. He is very hungry and very motivated to do it."
West Ham, who lost their first three Premier League games, stopped the rot with a 2-0 win over Huddersfield Town on Monday.
But the trip to The Hawthorns is another big test for Bilic and co, and another defeat would pile the pressure on the former Croatia boss whose days might still be numbered.