It could have been even worse for the Toffees in the evening Premier League clash.
In the dying minutes, 18 year-old Freddie Sears saw his stabbed shot hit a post and bounce along the line... and that is how close the Hammers came to a deserved victory.
Everton had started with power and pace, Ayegbeni Yakubu scoring his 19th goal of the season after eight minutes. West Ham were poor at that stage, and it seemed Everton would cruise it.
But West Ham, who had hit the bar through debutant James Tomkins, were the better side after the break.
Everton find themselves two points behind fourth-placed Liverpool, who can have no greater spur to win at Manchester United on Sunday than this misfortune to their neighbours.
West Ham have responded well to three 4-0 defeats in a week - with decent results against two sides above them in the table, beating Blackburn last week and now a point at Goodison.
The Hammers gave centre-back Tomkins his first start, and it was something of a baptism of fire against the power of Yakubu - even if the England Under-19s international could have scored inside five minutes.
For Everton, Tim Cahill returned after a hip injury - while teenager Victor Anichebe had also recovered from a similar injury to make only his ninth start of the campaign in place of hamstring victim Andrew Johnson.
Leighton Baines played on the left of midfield in the absence of Steven Pienaar, out with a muscle problem.
But it was Basildon-born Tomkins who almost made it a dream debut. He met Mark Noble's corner 12 yards out, and his looping header clattered against the bar.
But three minutes later, after Anton Ferdinand had lost out in the air to Anichebe, Tomkins was equally at fault when he was out-muscled by Yakubu midway inside West Ham's half.
The big Nigerian surged away and scored past Robert Green.
Cahill lasted just two more minutes, before suffering a recurrence of his injury. Manuel Fernandes replaced him.
Everton were searching for the second against a tepid, uncertain West Ham - and Yakubu was wrongly adjudged offside when he scored from just inside the box, the final pass actually coming off West Ham midfielder Scott Parker.
Everton had plenty of the ball and had West Ham on the back foot, with Anichebe and Yakubu causing problems.
But the Hammers had their chances.
Noble's low cross was mis-kicked by Mikel Arteta; the ball fell invitingly for Ashton on the six-yard line, but his shot went agonisingly the wrong side of Tim Howard's right-hand post.
Green made a brave save at the feet of Anichebe, when Yakubu had played him through four minutes from the break - but again West Ham could have equalised soon afterwards.
Ashton struck a low free-kick from 20 yards - and even though it deflected wickedly off Joleon Lescott, Howard made a remarkable save with his legs while going the wrong way.
Five minutes into the second period, Sears came on for Luis Boa Morte and was soon scampering around and unsettling defenders.
Everton still looked the most likely to get the next goal, though - and when Baines produced a run and cross, it took a fine Green save to halt Yakubu.
Ashton drove wide, and then Noble struck a free-kick over the bar.
Lee Carsley responded with a 25-yarder, when given too much space, but the effort flashed over. Then Green saved at Baines' feet after 65 minutes.
But three minutes later, West Ham got the goal.
Lucas Neill fired over a cross from the right, and Ashton rose above Phil Jagielka to send his header in off a post.
Freddie Ljungberg's pass then sent Sears in behind Lescott, but his shot from a tight angle was held by Howard.
After 80 minutes West Ham sent on Nolberto Solano for Ljungberg; then Spector replaced Parker.
When Sears robbed Jagielka on the edge of the box the teenager was on his way round Howard before the American goalkeeper just grabbed the ball.
Noble drove inches over the bar as West Ham put a fine passing move together; then Sears got away from Jagielka again, and this time saw his stabbed shot hit a post.
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