Arnautovic hit the winner on the day.
Pellegrini told whufc.com, “I can't say I've been impressed by Arnautovic, because I know him. For the three years I was at Manchester City I played a lot of times against him and he was always a good player who could do different things.
“Maybe last season was his best season in England, especially in the second half of the season. He demonstrated how important he can be. He's always running, he supports the ball and he scores. If he can repeat the second half that he had last season, he will be a top player."
Pellegrini also offered praise for the fantastic West Ham supporters who travelled to Adams Park for Saturday's contest, saying the team would not be the Hammers without their loyal following.
Pellegrini added: “I will say the same thing I said about Arnautovic about the West Ham fans. I cannot be impressed by West Ham fans because I know that over the years, the way they have supported their team has been unbelievable. They are one of the best groups of fans in the Premier League.
"West Ham moved to the London Stadium, which is not easy for a team. You go from 30,000 people to a stadium with 60,000 people, and it's full in the same way! One of the most important things this team has is their fans."