In a game of few clear-cut chances, defender Ogbonna netted the only goal in the 74th minute, rising inside the six-yard box to flick home Jarrod Bowen's corner.
The Hammers dominated the first half but lacked incisiveness in the final third and called Jordan Pickford into action just once.
The England keeper palmed out Bowen's stinging shot while Tomas Soucek thought he had found the breakthrough with his follow-up from close range but it was ruled out for offside.
The Toffees could not find an equaliser as their three-game winning run at Goodison Park was ended.
West Ham's victory takes them up to sixth in the Premier League table as they leapfrog Everton, who drop to seventh.
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