Yarmolenko celebrated his first league start for the Hammers with two gvoals before half-time as the pointless London club got off the mark at last.
The Ukranian has been left frustrated by his lack of opportunities so far since his £17.5million move to the London Stadium from Borussia Dortmund in the summer.
But he made his point in the best way possible, first opening the scoring on 11 minutes when the Hammers broke on the counter-attack for Arnautovic to feed his team-mate to fire past Jordan Pickford.
And nine minutes later he was at it again, but this time showing the class in his left foot to cut inside two Everton challenges before curling a shot into the far corner.
Gylfi Sigurddson gave Everton hope of a comeback when he powered home a header in first half stoppage time.
Marco Silva's side had already seen Cenk Tosun waste two good opportunities to score when Sigurdsson climbed above his marker to bullet a header into the far corner.
That should have been the perfect springboard for the Toffees to come out and have a real go at a Hammers side who have been leaking goals for fun this season.
But West Ham made certain of the points when Mark Arnautovic combined well with Pedro Obiang before sliding home the third.
But there was still time for Everton to create, and miss, a couple more chances as their lack of quality at the vital moment came back to haunt them.
Oumar Niasse and Tosun were both both guilty of squandering opportunities to set of a nervous finish and ultimately that cost them dear.
In the end West Ham were able to see out the rest of the game comfortably and were fully deserving of the three points which is enough to see them climb up to 16th in the Premier League.
But Toffees boss Silva will have to consider what he can do to change things up after suffering hif first defeat since taking charge – although the team have only won once in their five matches.
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