The victory was West Ham's first of the season in the league and lifted them off the bottom of the table and into 16th place, but Pellegrini said Arnautovic is in a race to get fit for Sunday's home match.
"Marko Arnautovic finished the match against Everton with a problem behind his knee," the manager told a news conference on Thursday. "We will see if he is fit for Sunday. He played well and we hope he will recover."
Fellow forward Javier Hernandez has been ruled out due to illness while target man Andy Carroll is not fit either after an ankle injury in pre-season, leaving Pellegrini with Andriy Yarmolenko and Lucas Perez as his only senior strikers.
Yarmolenko, who was signed from Borussia Dortmund in the close season, marked his first league start for West Ham against Everton with two goals, and Pellegrini said he was not surprised at the striker's quick adaptation to the English game.
"I said before that it would take Yarmolenko a few weeks to get used to the Premier League," Pellegrini added. "He has a lot of quality and he demonstrated that. This performance showed he is a top player."
Chelsea have made a perfect start to the season, winning all five of their games, but Pellegrini urged his players not to be intimidated by the form of the league leaders.
"It's impossible to start better than Chelsea have, but every game is different," he said. "We must be confident at home. We will try to do the same as the last game. We have to concentrate defensively, but I am confident."
(Reporting by Simon Jennings in Bengaluru, editing by Pritha Sarkar)
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