Lukaku’s first goal in seven matches ended a four-game losing streak for Roberto Martinez’s side and ensured they would live to fight another day in the competition after James Collins’ 56th-minute header had exposed the Toffees’ defensive problems.
Collins evaded the clutches of several Everton defenders to meet Morgan Amalfitano’s whipped near-post corner for what looked like being the winner before Lukaku’s late intervention,
The Hammers made the brighter start to proceedings but ultimately conceded the better chances to their hosts in the first half before grabbing the initiative midway through the second period.
For all of their endeavour, though, Everton squandered numerous chances to level the match with Lukaku the main culprit on several occasions.
The Hammers looked destined to have booked a trip to either Doncaster or Bristol City in the fourth round but Lukaku’s smart turn and finish in second-half stoppage time eased some of the mounting pressure on his manager Roberto Martinez and set up a replay at the Boleyn Ground next Tuesday.
West Ham were on the verge of their first victory in 15 attempts going back to 2007 against the Toffees but the Merseysiders were persistent in their pursuit of an equaliser and were rewarded for their efforts.
Sam Allardyce, who was without Cheikhou Kouyate and Alex Song, enjoyed a better start to the game but afforded their hosts the clearer openings.
Lukaku returned to the Everton starting line-up having been dropped to the bench for the last two games and it looked like the rest had done him some good as he twice forced Adrian to fumble shots in the first half.
The West Ham goalkeeper, however, recovered once to deny Kevin Mirallas on the rebound and was then bailed out by Winston Reid.
The Belgium striker also rolled an effort wide of the far post after he was picked out in the inside-left channel by Ross Barkley.
Steven Naismith came within a whisker of opening the scoring after he latched onto Lukaku’s chipped cross to flash a shot across the face of goal which ultimately went out for a throw-in.
Seamus Coleman and Mirallas both went close with decent attempts before Barkley, who had an inconsistent evening under the floodlights, showed his best and worst with a 60-yard drive at the West Ham defence that ended with him passing tamely into the path of Reid as he tried to pick out Lukaku.
The visitors were not without their chances with Stewart Downing forcing Joel Robles to tip over and Amalfitano firing wide before the break.
Everton’s opportunities began to dry up in the second half as, for all Lukaku’s good early play, the side were lacking runners to get beyond him to open up the Hammers defence.
The visitors seized their chance to grab the initiative when Collins rose at the near post, unchallenged, to fire a header past the despairing Robles to give the visitors the lead.
The Everton goalkeeper then reacted well to tip over Enner Valencia’s header minutes later while Lukaku’s own header was ruled out after referee Anthony Taylor adjudged him to have fouled Aaron Cresswell in the process.
Martinez had labelled the match as the most important of their season, and Lukaku pounced on Bryan Oviedo’s cross late on to salvage a replay from the jaws of an early cup exit could prove to be a turning point in the Toffees stuttering season.
Everton: Robles (6), Coleman (6), Jaielka (5), Distin (5), Oviedo (6), Besic (5), Barry (6), Naismith (6), Barkley (5), Mirallas (7), Lukaku (8).
Used subs: Eto’o (4), McGeady (5), Stones (5).
West Ham: Adrian (6), Jenkinson (6), Reid (6), Collins (6), Cresswell (6), Noble (5), Nolan (5), Amalfitano (7), Downing (7), Jarvis (6), Valencia (6).
Used subs: O’Brien (5), Poyet (4), Cole (4).
Man of the Match: Romelu Lukaku.