The result has left the Hammers in 14th spot in the Premier League table, boasting 12 points from their opening 11 matches, while Everton remain 16th, now on 10 points from their first 11 games of 2024-25.
Crysencio Summerville produced a wonderful goal-saving challenge to prevent Everton from scoring in the early stages of the contest after a quick counter-attack from the Toffees had ended with Abdoulaye Doucoure in a shooting position.
The fact that was one of the standout moments of the first period said a lot about the quality on show at the London Stadium, with both teams finding it difficult to make their mark in the final third of the field.
Everton did offer more in the first half, with West Ham goalkeeper Lukasz Fabianski needing to keep out a header from Dominic Calvert-Lewin, but Toffees stopper Jordan Pickford tipped a Jarrod Bowen effort wide of the post shortly before the break.
The visitors had a golden chance to score early in the second period, but Jesper Lindstrom headed wide of the post, before Pickford kept out an effort from Guido Rodriguez, as the tempo was lifted after a poor first period.
Fabianski was called into action just past the hour to deny Lindstrom, before Summerville hit the woodwork down the other end, with the two teams entering the final 15 minutes of the contest on level terms.
West Ham did look the more likely in the latter stages, with their best chance falling to Danny Ings, but the striker's attempt was kept out by Pickford, and it finished all square in the English capital.
MAN OF THE MATCH - JAMES TARKOWSKI
This was not a match for the attacking players, with almost all of them, potentially bar Summerville, finding it incredibly difficult to make an impact in what was a low-key game in the English capital.
The man-of-the-match award could have gone to either of Everton's centre-backs, with Jarrad Branthwaite also impressing once again, further boosting his growing reputation, but James Tarkowski won four aerial duels and made two tackles, while he completed 40 of his 48 passes during a strong display.
WEST HAM VS. EVERTON MATCH STATS
Possession: West Ham 49%-51% Everton
Shots: West Ham 11-18 Everton
Shots on target: West Ham 6-4 Everton
Corners: West Ham 7-6 Everton
Fouls: West Ham 10-9 Everton
BEST STAT
WHAT NEXT?
West Ham will be bidding to return to winning ways in the Premier League when they head to Newcastle United after the international break, before finishing the month at home to Arsenal.
Everton, meanwhile, will welcome Brentford to Goodison Park in their first match after the November international break, before visiting Manchester United on December 1.