Tony Hibbert's second own-goal at the Boleyn Ground in as many seasons helped West Ham United extend their unbeaten Barclays Premier League run to three matches.
The Everton defender inadvertently diverted Radoslav Kovac's overhead kick past a helpless Tim Howard to give the Hammers an early lead on Tuesday evening.
West Ham went into the tea-time kick-off full of confidence following a confidence-boosting 3-1 victory at Fulham on Boxing Day.
Hibbert's second piece of east London misfortune appeared as if it could spur West Ham on to their first back-to-back league wins since February, only for Irish youngster Seamus Coleman to equaliser with a well-taken goal three minutes before half-time.
The Toffees, whose own 26 December fixture at home to Birmingham City was postponed because of frozen pipes at Goodison Park, were much the fresher of the two sides and pressed hard for a winner in the closing stages.
However, it was West Ham's own Boxing Day hero, Carlton Cole, who had the clearest opportunity to win all three points for his side, only to drag his shot wide from Freddie Sears' pinpoint low cross.
Mindful of his players having completed a gruelling 90 minutes at Craven Cottage just 53-and-a-half hours earlier, Hammers manager Avram Grant made four changes to his starting XI.
Out went two-goal Cole, Danny Gabbidon, Junior Stanislas and Tal Ben Haim, with Victor Obinna, Herita Ilunga, Luis Boa Morte and Julien Faubert coming in to replace the quartet.
Everton defender Sylvain Distin had the game's first chance, only to drag his shot horribly wide after West Ham had failed to deal properly with a corner.
Hibbert made his central defensive partner pay for his profligacy a few moments later when this time Everton were unable to clear a corner from their own penalty area.
Scott Parker - typically immense throughout - sent over a dangerous corner which James Tomkins headed goalwards. The No5 then made a nuisance of himself as Tim Howard parried the ball and when Kovac acrobatically diverted the ball goalwards, it hit Hibbert's thigh and flew past the United States goalkeeper.
West Ham took confidence from the goal, with both Matthew Upson and Frederic Piquionne going close before left-back Ilunga was forced off with an injury.
Jonathan Spector was brought on in his place but when the American was dragged inside by an Everton player, Seamus Coleman ghosted in at the far post to smash Tim Cahill's cross past Robert Green.
There was still time for Parker to float a chip on to the roof of Howard's net before an entertaining first half was brought to a close by 2010 FIFA World Cup final referee Howard Webb.
Into the second period and it was Everton who spent most of the 45 minutes on the front foot as their hosts visibly tired following their Boxing Day exertions.
South Africa midfielder Steven Pienaar twice curled efforts just off-target, while Green had to be at his alert best to push away a powerful Marouane Fellaini header.
When West Ham did threaten, it was mainly on the counter-attack, and one such break almost brought a winner for Cole with 20 minutes remaining.
Obinna sent Freddie Sears away down the right but when the youngster sent over a low cross, the England forward could only roll his shot wide from 12 yards.
Parker admitted the result had left he and his team-mates feeling mixed emotions.
"Obviously after going one-nil up, we are a little bit disappointed. We needed to win but, in saying that, we have come off a good result against Fulham and we have been unbeaten. We have not lost the game."