Everton went close late in the half when Alex Iwobi’s goal-bound effort was deflected onto the woodwork by Declan Rice, and they were much improved in the second half but could not find a way back into the game.
West Ham sat back after the break, looking to absorb the visitors’ pressure and counterattack them swiftly.
From one of them, Everton goalkeeper Jordan Pickford pushed an Emerson shot over the crossbar, the closest either side came to a goal in the second period.
It spelled more sorrow for Everton fans, who staged a sit-in at Goodison Park last weekend demanding the club’s board be sacked following a defeat to Southampton. These calls may intensify in the coming days.
West Ham are next in action in the fourth round of the FA Cup when they travel to Derby County on 30 January, while Everton host Arsenal in the Premier League on 4 February.