The Hammers, who climbed off the foot of the table thanks to Sunday's victory at Fulham, took the lead when Tony Hibbert diverted Radoslav Kovac's tame overhead kick into his own net.
The visitors hit back three minutes before half-time through Seamus Coleman, although Carlton Cole wasted a glorious opportunity to win it for the home side.
West Ham enjoyed more than one slice of luck in their 3-1 win at Fulham on Sunday and they got another to take a 16th-minute lead. Tim Howard parried a James Tomkins header from a corner to Kovac, whose overhead kick looked harmless but took a huge deflection off Hibbert and into the net.
Everton began to get back into it and the home side suffered a blow when they were forced to replace injured left-back Ilunga with Jonathan Spector in the 34th minute. It proved costly as Everton equalised eight minutes later when Seamus Coleman got in behind the substitute to sidefoot home Tim Cahill's far-post cross.
Everton had several chances in the space of three minute to score a late winner. First Cahill steered Steven Pienaar's cutback straight at Robert Green, Scott Parker then produced a last-gasp tackle to thwart Yakubu before Green repelled Marouane Fellaini's bullet header from the resulting corner. Cahill did then slide the ball into the net but he was well offside after being found by Yakubu's mishit scissors kick.
Source: PA
Source: PA