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.
It was West Ham who showed precision in the 31st minute as they surged ahead, although a large deflection off Phil Jagielka helped them take the lead.
Matt Jarvis, once again proving a handful down the left wing, skillfully threaded his way infield and teed up Morrison and the former Manchester United midfielder drilled the ball home via Naismith.
Barkley was booked for a nasty tackle on Morrison that saw him stamp down his outstretched leg and Kevin Nolan attempted to overpower Tim Howard at the ensuing free-kick, but the Everton keeper showed courage.
Lukaku came on for the second half and his delicate early touch was taken on by compatriot Mirallas, who raced to the byline and crossed only for West Ham to recover by getting numbers back.
Naismith had also made way for McCarthy as Roberto Martinez attempted to invigorate his side, but the Hammers were more creative with Jarvis galloping free and just failing to find Modibo Maiga.
Showing vision once more, Lukaku played Barkley into the area only for England midfielder to run down a blind alley.
Everton maintained the pressure, however, with Baines sending a cross to the far post but it was too fast for Mirallas to reach.
The initiative had been seized and in the 62nd minute the equaliser arrived, Baines steering a terrific 20-yard free-kick beyond Jaaskelainen into the top left corner.
Noble flashed a corner wide of the left post and Baines forced a save from close range as the match opened up.
Noble powered home from the spot after being hacked down by McCarthy, but then he trudged off after receiving a second yellow from Mason and Baines delivered the equaliser.
The outstanding Lukaku deserved a goal and it duly arrived as he nodded Everton to victory.