A dreadful goal keeping error by Shaka Hislop gift-wrapped the three points, but Everton showed enough against an out-of-sorts West Ham to indicate that their elevated status could be sustained.
Francis Jeffers was the man to benefit from Hislop's slip, when the teenager's innocuous shot squirmed through the grasp of the goalkeeper, who was playing with a thigh strain. And Jeffers could have put the victory beyond doubt had he not needlessly strayed offside in the dying minutes before finishing a swift Everton break.
The goal released the shackles of caution that had, until that point, held Everton back. With West Ham largely dominant throughout and League debutant Abel Xavier enjoying a competent and unflustered midfield role, Hislop remained redundant for long periods.
But after the goal, and for the final quarter of the game, Everton were then able to push West Ham on to the back foot, mainly through the kind of forceful running from Nicky Barmby that set up the score.
When his route to goal was blocked, Jeffers took over and, as a crack appeared in the massed ranks of the visitors' defence, he did not hesitate to chance his arm.
"These are giddy heights," smiled Smith. "We have beaten teams that are expected to be around the same area as Everton, but West Ham would have looked at being a top six finisher this season, so I'm really pleased we got that win. I didn't think it was the best we have played, but the conditions were awkward," he added.
The West Ham manager, Harry Redknapp, attempted to remain calm. "Shaka has been fantastic and that was most unusual. He's entitled to the odd mistake once or twice a season. It had looked a banker 0-0 to me."
Both sides took time to adapt to a pitch drenched by persistent rain which contributed to a first half riddled with errors. The worst, when Everton defender David Weir misjudged the bounce of a clearance, allowed Paulo Wanchope in for the best opportunity, but Paul Gerrard used the conditions to his advantage, sliding out to smother the stretching attempt.
Smith was twice forced to re-shuffle his midfield when both David Unsworth and Don Hutchison were substituted with shoulder and calf injuries. Neither are expected to play against Oxford on Wednesday.
Goal: Jeffers (64).
Everton (4-4-2): Gerrard; Dunne, Weir, Gough, Unsworth (Gemmill, 45); Hutchison (Ball, 39), Abel Xavier, Collins, Barmby; Jeffers (Clelland, 90), Campbell. Substitutes not used: Cadamarteri, Simonsen (gk)
West Ham (5-3-2): Hislop; Sinclair, Potts, Stimac, Margas, Keller; Lampard, Lomas, Moncur; Wanchope, Di Canio. Substitutes not used: Forrest (gk), Kitson, Foe, Carrick, Newton.
Referee: S Bennett (Kent)
Bookings: Everton: Dunne, Barmby. West Ham: Stimac, Di Canio, Lampard
Man of the match: Dunne
Attendance: 35,154.Reuse content