The Toffees and the Hammers went into the game at Goodison Park having both accumulated just four points from their opening six matches of the season.
However, Everton were still waiting for their first maximum haul of 2022-23, instead being reliant on a run of four successive draws to avoid being cut adrift in the relegation zone.
Frank Lampard's side had impressed in the Meresyside derby two weeks ago, holding neighbours Liverpool to a goalless draw to give them confidence ahead of this fixture.
Although Everton did not replicate the standard of that performance, they were still able to get over the line courtesy of Neal Maupay's goal early in the second half.
Alex Iwobi fizzed a pass into the feet of the summer arrival, who was waiting on the edge of the area, and the Frenchman turned sharply before lashing a volley into the bottom corner from 20 yards.
Commenting on the goal, Maupay told Sky Sports: "Always the first touch is the key [for his goal]. We've got very good players in midfield and the wings. They make the difference. I have to be there to finish. I'm just happy for the team. It's been coming. We had a very good week of training and we deserved to win.
"I wouldn't say I feel under pressure. I know my job I have to score. That's why they brought me in. I needed that goal as I haven't played much last month.
"I wasn't playing for Brighton towards the end. It was only my second game of the season. I needed this goal. I'm here to help. I trust my teammates to put me in the right position to score."
For a player who had gone 12 matches without finding the back of the net, it was a fine finish not in keeping with a match which was often back-and-forth with little end product.
West Ham had also posed a threat with Said Benrahma going closest with a deflected effort from distance which struck the inside of the post and bounced clear.
Second-choice goalkeeper Asmir Begovic deputised for Jordan Pickford between the sticks and helped Everton keep successive clean sheets in the Premier League for the first time since May 2021.
Everton now sit in 13th position in the Premier League standings, while West Ham will return from the international break sitting in 18th place.