Maupay, a recent signing from Brighton & Hove Albion, took an excellent touch from Alex Iwobi’s pass and volleyed into the back of the net from the edge of the penalty area 53 minutes into the game.
The win ends a run of four consecutive draws for Everton and takes them to 13th in the standings, while West Ham remain in the bottom three with one win out of their seven games, their 1,000th loss in the Premier League.
Both teams blew and puffed through the first half without creating many chances. Everton had most of the ball, but failed to get a single shot on target.
West Ham created a succession of corners, some of which led to header opportunities, but none had the strength or direction to defeat home goalkeeper Asmir Begovic, who was standing in the wing for the injured Jordan Pickford.
Maupay’s 53rd minute goal relieved the home fans, but West Ham also sprang into action, hitting the woodwork via substitute Said Benrahma, whose curling shot Begovic failed to beat the post.