Dominic Calvert-Lewin wrote another chapter in his breathtaking start to this season with a hat-trick as Everton won their sixth straight game from the start of a campaign for the first time in 82 years.
This wasn’t as straight forward for Everton as the scoreline suggests.
Indeed the Blues were at their most fluent before half-time and were unlucky to lead only 1-0 at the interval.
Robert Snodgrass equalised immediately after the restart – which angered Everton.
They’ve been here before of late, pegged back by both West Bromwich Albion and Crystal Palace before running out winners.
Everton promptly hammered on West Ham’s door and crashed through when Richarlison meandered across the edge of the box before unleashing a drive which deflected off Declan Rice and wrong-footed Darren Randolph.
Richarlison was soon forced off injured, however and Everton will be keeping their fingers crossed following problems for three players.
The brilliant pair of Richarlison and Allan went off inside eight hurtful second-half minutes – after Jonjoe Kenny’s enforced withdrawal four minutes before the break.
It was Richarlison’s replacement Alex Iwobi who did a lot of the leg work for Everton’s third goal.
Collecting a pass from Gylfi Sigurdsson, he crafted room to shoot, striking the far post.
The ball rebounded for Calvert-Lewin to turn home.
And when Calvert-Lewin completed his second treble in 11 days, Everton were guaranteed their Carabao Cup fourth-round place.
Sigurdsson was the provider, forcing a pass through a narrow gap. Calvert-Lewin prodded past Randolph for his eighth goal of the campaign.
From Michael Keane’s carefully lifted pass from his own half, through Calvert-Lewin’s gorgeous touch to pull the ball from the sky, to the striker’s classy finish, Everton’s first goal was a work of art.
Calvert-Lewin left Declan Rice trailing to fasten onto the pass from Keane, who skilfully avoided a challenge from Manuel Lanzini before releasing his teammate.
The Everton forward used his left instep to control, steadying himself before easing his right-footed finish past Randolph.
Carlo Ancelotti’s teamsheet read like a statement of intent, James Rodriguez and Allan both included for their first Carabao Cup appearances.
And James was in sumptuous touch, gliding between wings to feed a succession of wonderful passes to his forwards.
One ball inside Ben Johnson for Richarlison defied science and drew audible gasps from those fortunate enough to be inside Goodison Park.
Richarlison gave it the finish it deserved, too, but the Brazilian had strayed marginally offside before placing the ball in the net.
James was having fun running with the ball, his opponents backing off every time.
Snodgrass got in a tangle and ended up on the deck trying to halt the Colombian in one instance.
When James went deeper, he fed a ball down the line for Calvert-Lewin – repeatedly running down young right-back Johnson’s channel.
Calvert-Lewin cut inside but sent his strike from 18 yards whizzing over the top.
For all the magical moments from James and Richarlison, it was Everton’s third South American in Allan who drew an appreciative roar from his watching teammates with a terrific challenge to thwart Felipe Anderson on the break.
Anderson had forced Jordan Pickford into a save down to his right on seven minutes but there was little from West Ham going forwards.
Lanzini tumbled in the box but was penalised for handball and Sebastien Haller couldn’t stretch far enough to meet a terrific Aaron Creswell delivery from the left.
Another Creswell cross, moments before half-time, was headed beyond the far post by Lanzini.
The visitors were unsure at the back, Creswell keeping Randolph honest with a hard-hit backpass and Fabian Balbuena heaving a sigh of relief when his gift of possession to Calvert-Lewin came to nothing.
West Ham’s hesitancy was understandable nevertheless. If it wasn’t James gunning for them, it was Richarlison.
Scorer of three goals in his past two games – including a double in the previous round at Fleetwood Town – Richarlison began here with a surge into the box which required three defenders to stop.
He danced through again on 20 minutes, slipping inside Balbuena and opting to shoot early with the outside of his left boot.
Randolph did very well to scoop the ball over at close range.
The one disappointment for Everton in the opening 45 minutes came with the injury to Kenny, who insisted on staying on the pitch to avoid leaving his team a man short defending a corner before hobbling off.
The home team sustained another blow scarcely 20 seconds after the restart.
West Ham’s equaliser was simple in its execution but an excellent strike from Snodgrass.
Anderson collected an overhit cross on the left and fed the ball back for Snodgrass on the edge of the box.
The Scot met it first time with a shot which arced across Pickford and into the right corner.
Everton responded with a Richarlison strike from the left of the box against the base of Randolph’s right-post and a shot from distance hit too high by Calvert-Lewin.
Allan was brilliant again to scupper Anderson on the counter, then it was James and Richarlison back to the fore, combining to supply Gylfi Sigurdsson in the middle of the penalty area.
Sigurdsson’s shot was firm but repelled by the excellent Randolph.
The goalkeeper was stranded soon after, though, Randolph initially shifting left when Richarlison let fly from 18 yards – but undone by a deflection off Rice’s back which sent the ball on a trajectory down the middle of goal.
Richarlison’s night was ended when he fell awkwardly, the South American replaced by Alex Iwobi.
The nature of these games, winner-takes-all and no extra-time meant West Ham would inevitably begin throwing caution to the wind.
A cluster of claret-and-blue shirts were waiting when Andriy Yarmolneko headed into the middle after Snodgrass dug out a cross.
The ball dropped for Lanzini but he dragged his effort wide.
Pickford athletically helped over an effort from Haller after the Frenchman controlled on his chest and aimed an overhead kick.
With Richarlison and Allan off, it appeared Everton were set for a backs-to-the wall job to see this out.
The red-hot Calvert-Lewin emphatically dismissed that notion.
Everton have scored 12 times in three games in this competition with a +9 goal difference.