The 'Hammers' were much improved from their recent displays as they looked to record their first points of the season, with a first-half brace from new boy Andriy Yarmolenko getting them off to the perfect start on Merseyside.
Whilst the first - laid on by Arnautovic - was a simple finish, the second was a stunner from the Ukranian as he curled a beautiful effort beyond Jordan Pickford.
However, Gylfi Sigurdsson struck on the stroke of the break to bring the hosts back into the game, making the next goal vital.
Despite Everton pressing significantly in search of the leveller it was West Ham who would get the all-important fourth goal as Arnautovic showed his predatory instincts to find the net.
The Austrian was also involved in the build-up, exchanging passes with Pedro Obiang before sliding home his return pass after spinning away from his marker on the edge of the box.
The goal was Arnautovic's third in five Premier League games this season for West Ham, with only he and Yarmolenko having found the net for the 'Hammers' so far this campaign.