Roberto Martinez suggested in the days ahead of this game that Romelu Lukaku is the complete striker.
It took the Belgian international, who is still only 20 years of age, approximately 21 minutes to provide compelling evidence to back up his manager's assertion in this match.
After more than a month on the sidelines through injury, the big hitman could have been forgiven for being a little rusty upon his return to first-team action.
Instead, he proved the difference on an afternoon when Sam Allardyce's well-organised and well-drilled outfit were proving an incredibly tough nut to crack, netting the only goal of the game eight minutes from time after arriving as a second-half substitute in the content.
Jose Mourinho will vouch for the Hammers' ability to make life hard for home teams when they are on top of their game.
Thankfully, the Chelsea loanee was able to prove he has that unerring quality all the best forwards have - the ability to be in precisely the right place at the right time.
Netting goals during periods of dominance has been a notable problem for the Blues during Lukaku's absence - as the defeats at Stamford Bridge and White Hart Lane illustrate.
Before Lukaku's arrival, Everton continued in that vein.
A bright, offensive opening from the home side which generated a swathe of possession garnered some tantalising crosses, a header wide from the effervescent Steven Naismith
and a bold drive from Barry that sailed wide. And all inside the opening 10 minutes.
Gerard Deulofeu, back in the starting XI for his first start since a hamstring injury dating back to before Christmas, was also catching the eye with some eager runs and neat link-up play. A sizzling solo run in the 39th minute was the highlight of his first-half display. Frustratingly for the home side, his shot at the end of it curled inches wide of the upright.
The visitors maintained a fleeting threat on the counter-attack, with the pace of Carlton Cole and tenacity of Kevin Nolan at the heart of their best moments in the opening half hour. But there was no doubt the home side were in command.
The arrival of former Liverpool striker Andy Carroll in place of Cole before half-time provided the visitors with a different kind of attacking threat.
A fierce near-post drive from Naismith in the 22nd minute served to illustrate that point.
But despite the dominance in the first period, the familiar tale of failing to convert and create chances - which has proved the side's biggest stumbling block in recent weeks - continued.
Defensively, in the absence of the ever-reliable Phil Jagielka
, John Stones
produced a display which belied his tender years - playing with a composure, confidence and assurance which suggests, yet again, that his future at the Club is very bright indeed.
The best chance of the opening hour of the game, though, came from a more established pairing. Leighton Baines
linked brilliantly with Steven Pienaar
down the left flank. The South African aimed a low shot towards the corner of the net but the faintest of touches diverted it onto the post. Deulofeu couldn't beat Adrian on the rebound.
Minutes later and referee John Moss waved away strong claims for a penalty by Pienaar after a collision with Downing and the South African saw another effort from a good position fly over the bar.
The slew of chances in quick succession led to an increase in the volume from the Goodison faithful.
The home side's attacking threat held far greater intent - and was more direct, although with the ball remaining predominantly on the deck.
By contrast, West Ham's best opening of the early stages of the second period came from a very direct route - a long ball to Carroll, who produced a well-struck drive that forced Tim Howard
into a finger-tip stop.
Nolan curled an effort wide from the left-edge of the area as the Hammers, who had been in such rich form coming into the game, continued to pose a sporadic threat.
With 30 minutes of the game remaining, Goodison said a big 'welcome back' to Lukaku. The imposing striker returned after a five-week absence through injury, replacing Leon Osman
. Baines took over the captaincy.
Deulofeu fired wide from the edge of a crowded area in the 63rd minute and Sylvain Distin
connected with a Baines free-kick on 68 minutes, only to send his effort wide.
Sam Allardyce introduced Mohamed Diame for Matt Jarvis as the game became more open and highly-charged. By way of response, Deulofeu was withdrawn in favour of the fresh legs of Aiden McGeady
as the Blues maintained their pressure on the visitors.
West Ham had chances - mainly from the edge of the box and off target. But, as the clock ticked down, it was the home side upping the ante and, equally, feeling the frustration at the continuing stalemate.
And so when the goal came, the release of joy inside Goodison was palpable. A typically adept period of controlled build-up play down the left including Baines and Pienaar culminated with a low, driven cross which was teed up perfectly for Lukaku from 10 yards. He side-footed confidently low into the net to the delight of fans and teammates alike.
After so many frustrating outings in recent weeks, there was no doubt Martinez's Blues were due an afternoon where their dominance was translated into the appropriate points return.
A player who has deserved rich praise for his endeavour and attitude in recent weeks is Naismith. And it was fitting to see the Scot receive a standing ovation when he was withdrawn with just a few minutes remaining and replaced by Ross Barkley
And so, with 11 games of the campaign left to play, the Blues remain very much in the hunt for European qualification.