The Hammers' teenage debutant James Tomkins almost made a dream start to his career in the fourth minute when he met Mark Noble's corner with a firm header only to hit the bar.
Sadly, his lack of experience was quickly exposed when he was turned by the wily Yakubu, who showed pace and power to race clear and fire an emphatic finish past West Ham keeper Robert Green.
Everton suffered a blow in the ninth minute when Tim Cahill, who had been doubtful before the game with a foot injury, suffered a recurrence and limped off to be replaced by Manuel Fernandes.
Yakubu should have had his second after 15 minutes when he turned in a neat finish. The goal was ruled out for offside, but replays showed the striker was onside and the final touch through to him appeared come from West Ham's Scott Parker.
West Ham have had had their struggles recently, but they responded well and should have been level by the interval.
A communication breakdown between Mikel Arteta and Phil Jagielka in the 31st minute gave Ashton the perfect opportunity right in front of goal but he somehow shot wide.
Yakubu was a real physical threat and Green needed to plunge at the feet of Victor Anichebe after he was played in by his strike partner.
Ashton was frustrated again two minutes before half-time when his 25-yard free-kick took a heavy deflection off Joleon Lescott, but Everton keeper Tim Howard reacted brilliantly to stretch out a foot to save.
West Ham clearly felt they were in with a real chance of drawing level, and boss Alan Curbishley wasted no time after the break in making a change, sending on teenager Sears for the dismal Luis Boa Morte.
And the Hammers were level after 68 minutes when Ashton rose superbly to head home Neill's cross - with Everton keeper Howard getting a hand to the ball but failing to keep it out.
A dreadful error by Jagielka almost let in Sears with six minutes to go, but Howard came to the defender's rescue with a vital interception.
West Ham were by far the better side in the closing stages, and midfield man Noble sent a fierce 30-yard drive just over the bar.
The lively Sears was too quick for Jagielka again in the dying seconds, but his shot rolled agonisingly against the post with Howard beaten.
Everton boss David Moyes: "I am a little frustrated - I thought we thought we did OK.
"We tried to get at them early but give them a little bit of credit because they came back well.
"West Ham are playing well under less pressure. We're in fifth and hoping teams don't catch us up. We're pleased with a point but should have three."
West Ham boss Alan Curbishley: "We came here today and I asked them to pass it around - we were disappointed we didn't take all three points.
"Freddie Sears came on and he gave our side a lift - he could have scored a goal late on.
"We have been on top in games but don't get enough goals to finish off teams."
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Interview: Everton boss David Moyes
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