Gianfranco Zola's side had looked all set for victory against the goal-shy Merseysiders after substitute Jack Collison had given the Hammers a deserved lead with his first senior goal for the club.
But somehow, West Ham saw three points turn to one and then none, as Joleon Lescott levelled before Louis Saha struck twice to keep Everton in seventh spot and leave the home side precariously in 13th place, just two points clear of the drop zone.
Last Saturday, Zola had arrested a forlorn four-match losing streak at Middlesbrough and, after claiming their first draw of the campaign down by the Riverside , he looked all set to bag a much-needed victory as this match entered its final stages.
Up until then, there only looked to be one winner and that was the Hammers, who had earlier made two changes with fit-again Scott Parker and Lee Bowyer coming in for substitutes Hayden Mullins and Collison.
Certainly, Parker wasted no time in marking his return to the side, when he unleashed a low, 18-yarder that skidded across the sodden Upton Park turf, before being tipped away at full stretch by Tim Howard.
And on the quarter-hour mark, Collison found himself back in the fold, too, when he stepped from the bench to replace Matthew Upson, who was worryingly stretchered away after landing awkwardly in an aerial challenge with Victor Anichebe.
Midway through the half, Freddie Sears superbly released Craig Bellamy but his shot was charged down and, shortly afterwards, the Welshman had another shot headed clear by Phil Jagielka, before volleying a third effort high into the damp East End air.
On the half-hour mark, the buzzing Bellamy brilliantly turned Joseph Yobo before cutting back to Sears, who sent a rising, ten-yarder crashing back off the crossbar and, as the half drew to a close, the teenager then drilled a low, angled shot behind.
With last weekend's match-winner Marouane Fellaini suspended, Everton had also kicked-off minus the injured Yakubu and Steven Pienaar following their win over Fulham - a situation that had forced David Moyes to bring in Saha, Anichebe and Jack Rodwell.
And apart from a handful of speculative first-half efforts from Saha, Tim Cahill, Mikel Arteta and a rising Leon Osman shot, the muted Merseysiders had shown little attacking appetite.
Having been on the back foot for those opening 45 minutes, it was not too difficult for Everton to look livelier after the break and, with the hour-mark approaching, young Rodwell sent his downward header bouncing just inches wide.
On 63 minutes, however, the Hammers finally broke the deadlock, when Parker back-heeled into the path of Collison and the young Welsh international calmly curled a 12-yarder beyond Howard's outstretched left-palm to give his side a merited lead.
In reply, Lescott agonisingly headed Jagielka's deep free-kick across the face of goal and, concerned by the fact that the Hammers defence that had somehow failed to keep a clean sheet in any of its previous 25 league and cup matches, that was enough to give the home fans amongst the crowd of 33,961 the jitters.
Certainly, their fears were well-founded, for Everton were about to embark on a remarkable rampage that would see them score those three goals inside five manic minutes.
On 83 minutes, Lescott stole in front of James Collins to nod home Saha's cross into the back of the net.
And then goal-maker turned goal-taker as the twisting and turning Frenchman collected from Anichebe sent a low, angled 15-yarder in off the sprawling Robert Green's right-hand post.
With an East End evacuation by the stunned Hammers fans now in full flow, that man Saha then lashed home his second goal, after Cahill intercepted Julien Faubert's woeful pass, before inviting his team-mate to fire a low 25-yarder beyond the home keeper and seal an improbable win for Moyes' men.