The Austrian, who signed for the club from Stoke City for £25m earlier this summer, was given his marching orders for an elbow on Southampton defender Jack Stephens.
The incident occurred 32 minutes into the eventful match, which ended in a 3-2 defeat for the Hammers at St Mary's.
Ahead of the East London outfit's EFL Cup game against Cheltenham Town, Bilic told reporters: "You can't do those kind of things. I am fuming, still now - you can feel. I am fuming, but it happened. But no more. Not him, not anybody else. I am confident the message got through.
"The players they know and they can't let the team down and also of course him. And whoever does those kind of things will get the fine, of course. He will get fined. He was angry at himself. He still is. He regrets it, so he is angry at himself, but he can't do it any more.
"We had a chat. He said he is sorry, and he apologised, I don't know, three or four times - on social media, here, in front of the whole team, on the bus, in the dressing room, everywhere. I said, 'Yes, it is nice.' Of course it is nice, but you can't do that. That's it.
"How can he explain that? You can't explain that. You can't say I was late or whatever. He didn't want to elbow him and he didn't elbow him like this, with a swinging arm. He wanted to probably get him somewhere here, in the side, to push him a bit because he was elbowed a few minutes before. But his problem is their player went a bit down so he got him here, in the neck which is already nasty. It doesn't look good."
Arnautovic will now serve a three-match ban.