With the teams locked at 2-2, Lukaku nodded home Kevin Mirallas' cross with six minutes left to celebrate his Toffees debut by plundering the decisive score.
The Belgium striker, whose half-time arrival at the expense of Nikica Jelavic galvanised Everton, needed treatment after clashing heads with Joey O'Brien in the act of scoring but was able to continue.
The dominant figure on the pitch, it was a superb performance from Lukaku that strengthens the suspicion Chelsea were foolish to send him to Goodison on a season-long loan.
The Toffees had equalised twice through Leighton Baines, who struck two brilliant free-kicks in front of England coach Roy Hodgson to keep his side in contention.
The second arrived in the 84th minute after Mark Noble was shown a second yellow card after hacking down Ross Barkley from behind.
It was cruel twist to the afternoon for Noble, who shortly before had dispatched a penalty he had won himself after having his legs swept beneath him by blundering substitute James McCarthy.
West Ham - missing attacking options Andy Carroll, Joe Cole and Stewart Downing to injury - can at least take comfort from the end of their scoring drought which had lasted 314 minutes until Ravel Morrison fired them ahead after half an hour.
Everton went close after six minutes when Mirallas tested Jussi Jaaskelainen and Steven Naismith was inches away from a second attempt when the ball sailed past.
There was plenty of early endeavour from both teams but the visitors were more precise in the final third with the busy Mirallas looking dangerous.
Source: PA
Source: PA