The Austrian striker, a £25m club-record signing from Stoke in the summer, had failed to find the net in his first 11 games for the club.
But he finally broke his duck six minutes into Saturday's derby clash with Chelsea and that, coupled with a fine defensive display from the home side, was enough to condemn Chelsea to their first defeat in seven league matches.
Moyes made a big call ahead of Chelsea's visit, dropping England goalkeeper Joe Hart.
But Hart's replacement, Spanish keeper Adrian, justified his selection with fine first- half saves from N'Golo Kante and Davide
Zappacosta.
Chelsea were carved open far too easily with West Ham's first attack, when Arnautovic played a quick one-two with Manuel Lanzini.
Arnautovic nipped in behind Andreas Christensen and, as Cesar Azpilicueta stood off, curled his shot around the defender and
keeper Thibaut Courtois.
Chelsea had plenty of time to respond, yet this was a strangely subdued display from a side with six wins from their last seven outings.
Alvaro Morata, often isolated in attack, cut a frustrated figure and was lucky to escape a booking for a blatant dive as he ran past
Winston Reid.
Moyes had said beforehand that he had no one capable of man-marking in-form Eden Hazard, but he gave Pedro Obiang the job anyway.
And the Spaniard stuck to his task so manfully that Hazard will probably still see him in his bathroom mirror when he has a
shave in the morning. West Ham had, of course, led Manchester City
at half-time last weekend only to leave the Etihad without a point.
Blues boss Antonio Conte threw on Pedro at the break for the ineffective Tiemoue Bakayoko to give his side more of an attacking edge.



But the anticipated onslaught never came, and it was West Ham who should have scored early in the second half when Antonio played Arnautovic through on goal.
However, this time the former Stoke winger placed his shot too close to Courtois.
The hosts called for a penalty moments later when the ball struck the arm of Christensen,
but referee Anthony Taylor shook his head.
Conte had used all three substitutions by the 64th minute, sending on Victor Moses and Willian in a desperate bid to retrieve the situation.
Zappacosta curled a shot narrowly wide from the corner of the area and Adrian dived gratefully on to a deflected effort from Moses.
But when Chelsea's clearest chance arrived,
six minutes from time, Morata snatched at his shot and clipped the sidenetting.
Instead West Ham held out for a vital win that boosts their prospects of escaping relegation trouble.