Arnautovic hit the back of the net early in the first half after a quick 1-2 pass at the edge of the box.
Chelsea fought hard to score but the Hammers put up a tough display to deny the Blues despite Morata missing two sitters.
Chelsea remain 3rd on the table with the 1-0 loss and could be displaced if Liverpool win tomorrow while West Ham remain in the relegation zone.
Watch full match highlights below SHARE THIS STORY USING ANY OF THE BUTTON BELOW ⬇ PLACE YOUR TEXT ADVERT BELOW ⬇⬇⬇
Family Reveals Natural Cure For Staphylococcus, Syphilis, Gonorrhea And Other Related Infections. Click Here To See How New Discovery! Miracle Device That Cures DIABETES, BLOOD PRESSURE, STROKE, ARTHRITIS, PAINS, OBESITY And 50 Other CHRONIC DISEASES Without Drugs Or Herbs