Jarrod Bowen registered West Ham's first attempt in the 44th minute
West Ham failed to ease the pressure on manager Julen Lopetegui as they played out an uneventful goalless draw at home to Everton.
The visitors settled more quickly at London Stadium but other than a tame Iliman Ndiaye effort from the corner of the penalty area, Hammers goalkeeper Lukasz Fabianski was untroubled in a dull first half.
West Ham, meanwhile, were virtually non-existent as an attacking force but almost took the lead on the stroke of half-time, when Jordan Pickford kept out Jarrod Bowen’s near-post drive with a one-handed stop.
Lopetegui's team were marginally improved after the restart and Guido Rodriguez forced another fine save out of Pickford 10 minutes into the half, before Fabianski tipped a goalbound Jesper Lindstrom header over the crossbar at the other end.
Crysencio Summerville went closest to a breakthrough with 17 minutes remaining, his low shot rebounding off the foot of the far post with Pickford beaten.
The Everton goalkeeper produced another marvellous stop in added time to keep out Danny Ings' deflected drive as Everton held on for a share of the spoils.
West Ham stay 14th in the Premier League table, two points and places above Sean Dyche's side.
Too little, too late from West Ham
Such was the Hammers' performance at Nottingham Forest last weekend that some fans may view a goalless draw at home to Everton as a baby step in the right direction.
That, however, would be a fairly generous assessment.
There was no lack of commitment from Lopetegui's players in the early stages of Saturday's contest, with both Aaron Wan-Bissaka and Summerville making superb recovery runs to deny Abdoulaye Doucoure a clear sight of goal, but the home side were horribly disjointed in the final third before the interval.
The home supporters had to wait until the 44th minute for their first attempt of the afternoon – a powerful Bowen shot that Pickford did well to divert behind for a corner.
West Ham 'have to do better' - Lopetegui
Lopetegui's team looked the more likely to snatch all three points after half-time but Summerville was denied by the post after a slaloming run, before Ings' low drive was parried by Pickford.
The Hammers substitute almost won it deep into stoppage time but was thwarted by another outstanding save by Pickford, who adjusted his body to tip the ball over the crossbar.
The final whistle was met by boos by the West Ham fans, whose side face Newcastle away and Arsenal at home in their next two league matches.
On an afternoon of few positives for the hosts, they did at least manage to keep a first home Premier league clean sheet since January – and only a second shutout in 29 top-flight encounters.
Toffees grateful to Pickford
Everton need to find 'a bit of devil' in attack - Dyche
Everton will certainly be the happier of the two teams after picking up a hard-earned point on their travels, but Dyche may feel his side could have emerged with more than just a draw.
The visitors worked themselves into some good positions in the first half in particular, but Ndiaye's low effort and a Dominic Calvert-Lewin header shortly before half-time failed to trouble Fabianski.
After edging a first half low on quality and cutting edge, Everton faded in the second, and had Pickford to thank for preserving the point after the England international denied Ings at the last.
Everton have now lost only one of their past seven league games after slipping to defeat in each of their first four matches of the campaign.
However, they have won only one of their past 17 on the road in the top flight, collecting just nine points in the process.
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