Stewart Downing’s strike was cancelled out by Leon Osman before Lukaku extended his remarkable record of six goals in six games for Everton against the Hammers with a stoppage-time header.
It could be next Sunday’s match at his former club Newcastle is Allardyce’s last as Hammers boss.
But the 60-year-old insisted he took no time to soak in his surroundings for possibly the final time with his contract set to expire and talks on hold until the end of the season.
“I’m frustrated with the team and the result. I’m not worried about my position,” Allardyce said.
“I’m 60 years old, I don’t have to worry about my position any more.
“At the moment I’m still here, I’m still going to get the team ready for next week.
“The owners decide how they want to go forward and I wait for them. It’s as simple as that.”
Allardyce pointedly described the contest as “entertaining” and talked up his record in four seasons as boss, which included earning promotion to the top-flight and steadying the Hammers in mid-table.
“I don’t think I need to prove what I can do as a manager,” he said.
“It’s four outstanding seasons. Only threatened with relegation since we’ve been in the Premier League for about six weeks, last December (2013) through to mid-January (2014).
“My average finish in my entire career in the Premier League is 10th. And finishing in the top half in about six seasons out of the 14 seasons I’ve been there.
“If you want to know what I do as a manager, have a look at that.”
Allardyce thought his side, who are now level with Everton in 10th with one game to go, had managed to stop Lukaku, but the Belgium striker popped up with the winner.
“Unbelievable. I thought we’d done it, finally at last,” Allardyce added.
“He hadn’t caused us that many problems, but finally he was there to finish us off.”