But Lukaku volleyed home after Bryan Oviedo crossed into the area to force a replay and relieve the tension and frustration that had been building inside Goodison Park as well as lifting some of the pressure off Everton manager Roberto Martinez.
It was a first goal in seven matches for Belgium striker Lukaku, who was signed for 28 million pounds ($42.42 million) from Chelsea in the close season but has struggled to justify the price tag in a mixed start to the campaign.
"It would have been a crime if we had been out of the cup," Everton boss Martinez told BT Sport.
"I am pleased with the character and the reaction we showed. We are low on confidence, but it was a phenomenal reaction and it could easily be a turning point."
Everton have won only one of their last 10 matches in all competitions and hopes that Lukaku's late leveller could be a catalyst for an upturn in fortunes will face a stern test when they host Manchester City in the Premier League on Saturday.
The result extends West Ham's winless streak to four games ahead of next week's replay at Upton Park with the winners facing a clash against Doncaster Rovers or Bristol City.
For West Ham manager Sam Allardyce, whose team's excellent start to the season has tapered away in recent weeks, conceding such a late goal was a bitter blow.
"We've had our pocket picked," he said. "We are disappointed to lose a goal in injury time. I'm satisfied with the overall effort of the players but if you don't punish your opponents when you have the chance it comes back to bite you."
The evening's other tie also finished level as Chesterfield fought back from 2-0 down to draw 2-2 with fellow third tier side Scunthorpe United. Derby County await the winners in the fourth round.
(Reporting by Toby Davis; editing by Ken Ferris)
