Hughes was unhappy with the Austria international's salute to the home supporters in response to taunts he was receiving from them.
Arnautovic, who spent four seasons at Stoke, was shouted at by Hughes as he was substituted off shortly after scoring the Hammers' second goal of the game.
"There was a little bit of baiting obviously, but did he not expect that?" Hughes said.
"I had better be careful because it sounds like sour grapes [but] I think he should have risen above any criticism or abuse that he had.
"He had enough at other places, and this place was actually very, very good for him. The fans loved him here. So to burn all your bridges after just one game, he didn't need to do it."