It hurts a lot – Arnautovic concerned over knee injury
Arnautovic played a vital role as the Hammers picked up their first points of the season at Goodison Park, setting up Andriy Yarmolenko for the opener before scoring West Ham's third himself.
But the Austria international had to be withdrawn shortly afterwards, having gone to ground clutching his left knee.
Although he was able to walk off, Arnautovic was not able to downplay the injury.
"I don't know, we need to check now," he told Sky Sports.
"It's not my calf, it's my knee. It hurts a lot. We will need to check this week."
West Ham's initial front three of Arnautovic, Yarmolenko and Felipe Anderson impressed, causing Everton no end of problems.
MASSIVE WIN!
A brace from Yarmolenko and one from Arnie give us the points!#EVEWHU #COYI pic.twitter.com/NdO4qQbv9t— West Ham United (@WestHamUtd) September 16, 2018
But Arnautovic was quick to highlight the collective display of West Ham's midfield, with Declan Rice and Pedro Obiang particularly effective.
"It's not about us three [the front three]," the forward said. "It starts from the goalkeeper [Lukasz Fabianski] and everyone has done an unbelievable job.
"The midfield was everywhere and then it's down to us to score goals and win games. Credit to Yarmolenko for his goals."
From Everton's perspective, however, Marco Silva feels his side were guilty of mistakes which made things easy for their visitors.
In the end, Silva was left ruing Everton being taught "a lesson".
"We weren't good enough to win the match," said the Toffees boss. "We started well, until the first mistake we made.
"We prepared for the situations when the opponent was on the ball, with Arnautovic, we prepared, but at that moment we did not do what we needed to do.
"We conceded too many chances defensively, but we created the chances to score, a good number of clear chances. It’s a big lesson for us to learn fast.
"It's the Premier League, it's the way. Of course you have to improve to see what we did wrong."