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.
Next Sunday’s match at former club Newcastle could be 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. “I’ve never heard them (the board) say I’m not doing a good job. I’ll find out next week and then I can ask them, and see if they give me an honest answer.”
Goalscorer Downing believes the Hammers would be in contention for a top six finish but for the concession of late goals.
“When you’ve been conceding late it could be on people’s minds, but you’ve got to block that out.
“If we eradicate those mistakes, individually and as a team, and with the addition of a few more quality players we can be a European place team. We’re not far away.”
For Everton, left-back Brendan Galloway hopes he is at the start of a promising career after a winning debut.
“I found out at 1.30pm when the manager named the team. I wasn’t expecting it,” said the 19-year-old, who replaced the injured Leighton Baines.
“I worked hard in training this week, but I didn’t expect to be given the nod, but when I did I was really happy and looking forward to it.
“I thought the team did really well. It was important to get the win and to do it at the end made it more special.”
Adrián, Jenkinson, Collins (Burke 14), Reid, Cresswell, Noble, Kouyaté, Song, Downing, Cole (de Carvalho 70), E Valencia
Howard, Coleman, Stones, Jagielka, Galloway, Barry, McCarthy, Lennon (McGeady 78), Barkley (Mirallas 70), Osman, Lukaku