The Belgium international struck in the first minute of injury time to save his side, acrobatically volleying past Adrian to send Goodison Park wild and ease the pressure on Roberto Martinez after a run of four consecutive losses.
Everton had dominated the first half but were unable to turn that into goals, with Lukaku pulling one effort wide and seeing Adrian keep out another.
The hosts were made to rue those missed opportunities when the unmarked James Collins rose to nod West Ham ahead four minutes before the hour.
However, Everton continued to push for an equaliser and were rewarded in the dying embers of the match as Lukaku turned in Bryan Oviedo's cross.
The two sides will meet again at Upton Park in the replay on Tuesday 13 for the right to face either Doncaster Rovers or Bristol City in round four.
Having experimented with a back three in recent weeks, Martinez opted for a flat-back four, with Oviedo one of three changes – the Costa Rican replacing Leighton Baines.
West Ham were forced into four changes due to the Africa Cup of Nations taking Diafra Sakho, Cheikhou Kouyate and Alex Song away, while the injured Andy Carroll was replaced by Enner Valencia.
A scrappy start meant it was not until the 13th minute that either goalkeeper was called into action.
Lukaku spun away from his marker on the edge of the penalty area and fired a low shot towards goal, Adrian fumbled the effort but recovered quickly before Kevin Mirallas could pounce.
Another two chances fell the hosts' way seven minutes later as Steven Naismith fired wide, before Mirallas saw his effort blocked after Gareth Barry had robbed Kevin Nolan.
Lukaku was next to be given space inside the West Ham half, the striker released by Ross Barkley, only for his low shot to creep wide of Adrian's post.
West Ham finished the half in the ascendancy and could have gone in ahead had Morgan Amalfitano not blazed his effort over the bar with two minutes remaining.
A marauding run from Carl Jenkinson five minutes after the break nearly sparked the game into life. Having started on the halfway line the Arsenal loanee surged into the penalty area, but his low shot was pushed away by Joel Robles.
They did not have to wait long to break the deadlock, though, as Collins capitalised on some slack marking to turn in right-wing Amalfitano's corner.
Collins found himself in action at the other end three minutes later as Lukaku's low shot look destined for the bottom corner, only for the defender to clear at the last moment.
Martinez's decision to introduce Eto'o in the 65th minute gave Everton a new dimension in attack, but still they struggled to find a way through.
Barkley and Lukaku led the Everton charge in the closing 10 minutes, and it was the latter who came to their rescue as he fired past Adrian in injury time - his eighth goal of the season in all competitions denying Sam Allardyce his first FA Cup win as West Ham boss.
In Tuesday's other third-round tie, Chesterfield came from 2-0 down to earn a 2-2 draw and a replay against Scunthorpe United. The winners will head to Derby County.
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