There was a large slice of good fortune about the 57th minute winner, when Joe Cole's shot deflected past keeper Steve Simonsen via the chest of Trevor Sinclair, but it was perhaps fair reward for the side that showed most ambition.
West Ham, with Fredi Kanoute shrugging off the heavy cold that ruled him out at Villa Park on Saturday to join Paolo Di Canio in attack, were unable to find their rhythm for long periods.
Glenn Roeder reverted to 4-4-2, with Sinclair asked to form an unfamiliar central midfield partnership with Cole, while Sebastian Schemmel and Vladimir Labant operated on the flanks. Sinclair had Hammers' best effort of a disappointing fi4rat half - a first time, right-footed strike from 25 yards that Simonsen held after neat build up from the front pair.
But while West Ham struggled to open up Everton's back four, the visitors posed little threat at the opposite end, where David James was untroubled. Everton's best attempt was a run by Jesper Blomqvist that ended with a weak shot that trickled wide.
Despite Jermain Defoe's lengthy warm-up on the touchline before and after the half-time interval, Roeder kept faith with his starting line-up and the goal was his reward.
Cole, receiving Labant's ball inside from the left, sidestepped his marker 20 yards from goal and unleashed a swerving right-foot shot that struck Sinclair and diverted past Simonsen at his near post.
West Ham should have put the game beyond Everton's reach after 75 minutes when Cole won possession inside the centre circle and ran at the heart of the Merseysiders' defence. But the England international delayed his through ball to Kanoute a fraction too long and, though the Frenchman collected the pass and rounded Simonsen, he was rightly flagged offside with the open goal at his mercy.
Di Canio, who never stopped running on a night when little he tried came off, picked out Kanoute with a delightful, diagonal floated pass, but Fredi blazed his right-foot volley into the sidenetting from a tight angle.
It was Kanoute's last act of a subdued performance before, with five minutes left, he made way for the energetic Defoe - not that the England Under-21 star had much chance to make an impact.
Five minutes earlier, Roeder's first change was to send on the 20-year-old Richard Garcia for Schemmel, which must have pleased his mother, Maria, who had flown in from Perth, Western Australia to see her son play for the first time this season.
David James was a virtual spectator in the second half and not even the late introduction of David Ginola - on up front for the injured Kevin Campbell - could inspire a dour Everton, who never looked like equalising.
Di Canio, who earned international acclaim for his sportsmanship at Goodison Park last season, took pity on the Everton again when Sinclair's promising surge towards the penalty area was halted by an injury to Alan Stubbs. While a section of the home crowd booed in cynical fashion as Sincs kicked the ball towards the touchline, the eccentric Italian pleaded for sympathy from the frustrated fans in the Bobby Moore Stand, before checking the condition of Stubbs, who was genuinely hurt.
With so many players either injured or playing out of position - as Roeder said: 'We had two right-backs and two left-backs out there tonight' - maybe it was unrealistic to have expected a performance of high quality and cohesion. At the end of the night, it's another three vital, hard earned home points, no matter how they were achieved.
The West Ham manager revealed afterwards that Don Hutchison - stretchered off in the last home game, against Middlesbrough, leaves for Canada tomorrow morning and will undergo an operation on his torn knee liganments this Friday.
6 - Sinclair and Di Canio combine well to find Schemmel on the right, but the Frenchman's cross is too weak to reach the danger area.
9 - Campbell's fierce left-foot shot from the edge of the area is blocked well by Repka.
16 - Di Canio's quick-thinking pass releases Kanoute on the counter-attack, but after beating Pistone on the halfway-line, the Frenchman is foiled by the sliding challenge of Carsley on the edge of the area.
17 - Winterburn's hopeful 30 yard effort sails a few feet over the bar.
18 - Sinclair finds Kanoute, who in turn feeds the overlapping Schemmel inside the area. The newly-converted winger takes a bad touch but then finds his feet to fire in a decent effort that is deflected over for a corner. From Di Canio's quick flag-kick, Labant shoots into the crowd and the danger is cleared by Everton.
23 - Pistone's long throw from the right is only cleared to the edge of the area, where Carsley is waiting to fire in a snapshot volley that James holds well.
24 - Blomqvist is allowed to cut in from the left and unleash a shot that flies high over the bar.
25 - Cole's lightning reactions from an Everton clearance allow Di Canio to gain possession but, in fear of an offside flag being raised, the Italian chooses not to release Kanoute, and instead goes on his own before being halted by Unsworth.
27 - A long clearance falls to Di Canio, but his left foot shot is easily gathered by Simonsen.
28 - Kanoute's pass finds Di Canio in the area, and this time the Italian's fierce right foot effort is deflected over for a corner.
31 - Sinclair collects a loose ball around 25 yards out and tests Simonsen with a blistering effort that the Everton keeper holds well.
32 - Pearce gets forward to good effect as he dribbles along the edge of the area and finds Di Canio, whose attempted chip isn't clever enough to trouble Simonsen.
38 - Alexandersson's chipped cross from the right finds Campbell unmarked in the area, but the Everton captain thankfully heads straight at James.
39 - Blomqvist booked for foul on Schemmel.
45 - Kanoute's clever pass almost releases Cole, but the ball just runs away from him and into the grateful arms of Simonsen.
Added time: 1 minute.
45 - Blomqvist is allowed time and space to measure a shot, but fires just wide of the post.
Half-time: West Ham United 0 Everton 0.
46 - Action straight away as Simonsen's clearance is charged down by Di Canio, but kicked into touch for a corner by Pistone before Kanoute can pounce.
53 - Good work from Cole and Schemmel sets Kanoute free in the box, but the Frenchman's touch is too strong and the ball runs out of play.
54 - Labant's corner is cleared to the edge of the area, where Di Canio is waiting to fire a volley straight into the arms of Simonsen.
57 - GOAL! Hammers get the goal their efforts deserve, as Cole's goalbound effort takes a ricochet off the chest of Sinclair and completely wrong foots Simonsen in the Everton goal to land in the bottom corner. West Ham United 1 Everton 0.
63 - Substitution for Everton, Ginola replaces Campbell.
64 - Hammers are denied what appears to be a clear penalty when Di Canio spins in the box and is sandwiched by two Everton defenders, but sees his claim waved away by referee Knight.
68 - Double substitution for Everton, Linderoth replaces Carsley and Gravesen replaces Alexandersson.
75 - Some excellent footwork from Cole sets the Hammers on the attack, but the England youngster delays his pass a second too long and Kanoute is flagged offside.
79 - Another scintillating run from Cole almost ends in glory as he jinks past three opponents but fires straight at Simonsen from the edge of the area.
81 - Substitution for West Ham, Garcia replaces Schemmel.
83 - Di Canio's flighted pass finds Kanoute, who takes a touch and sends in a powerful effort that crashes into the sidenetting.
84 - Substitution for West Ham, Defoe replaces Kanoute.
Added time: 3 minutes.
90 - Pistone booked for foul on Cole.
Full-time: West Ham United 1 Everton 0.
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