The Toffees enjoyed much of the early play but the Hammers went on to break the deadlock after just 11 minutes.
A long ball from the visitors took out the Everton defence and allowed Marko Arnautovic to slide the ball into the path of Yarmolenko to smash home from close range.
West Ham grew into the game and had several more chances to increase their tally, Mark Noble threatening with a cross to the back post and, two minutes later, Issa Diop forcing a save from Jordan Pickford.
Everton then spurned a great opportunity to equalise on 25 minutes when Lucas Digne clipped a ball for Cenk Tosun, who was unmarked six yards out, but the Turkey striker could only head straight at Lukasz Fabianski.
Marco Silva's charges would come to rue their wasted chances as they fell further behind just after the half-hour mark, Yarmolenko pouncing on Pickford's miscued clearance and curling home from the edge of the area.
To their credit, Everton fought back and managed to halve the deficit just before half time through Gylfi Sigurdsson, connecting perfectly with Jonjoe Kenny's cross to head beyond Fabianski in the Hammers net.
The restart saw the Merseysiders drive forward at every opportunity in a bid to draw level, but they were caught out once again when Pedro Obiang burst through the opposition lines on the hour and cut back for Arnautovic to toe-poke past Pickford.
West Ham had the woodwork to thank for a gilt-edged Toffees chance to go amiss when, just four minutes from time, Digne found Oumar Niasse unmarked six yards out, only for the Senegal international to volley against the bar.
From then on Manuel Pellegrini's side had little trouble seeing out the remainder of the game plus five minutes of stoppage time, leading to celebrations amid the Hammers contingent as the team finally got off the mark in terms of league points.
EVERTON (4-2-3-1): Pickford; Kenny, Holgate, Zouma, Digne; Gueye, Schneiderlin (Bernard 44'); Walcott (Lookman 77'), Sigurdsson, Calvert-Lewin; Tosun (Niasse 69')
WEST HAM UNITED (4-1-4-1): Fabianski; Zabaleta, Balbuena, Diop, Masuaku; Rice; Yarmolenko (Snodgrass 82'), Noble (71'), Obiang, Anderson; Arnautovic (Antonio 64')