The Hammers looked to be on the verge of progress to the Fourth Round Proper after James Collins had headed them in front at Goodison Park.
Everton 1-1 West Ham United
The FA Cup
Third Round Proper
Tuesday 6 January 2015
Goodison Park, Everton FC
Live on BT Sport
Winners receive £67,500
But after squandering further opportunities to double their lead, a late equaliser from Romelu Lukaku for the hosts rescued a 1-1 draw and the sides will now meet again at the Boleyn Ground next Tuesday.
"We did think we were there," admitted Allardyce.
"We were well worthy of the lead and the disappointing thing was that Enner [Valencia] didn't score at that crucial point.
"At this level of football, when you get that golden chance you can't afford to miss it.
"At 2-0 that kills Everton's spirit. At 1-0 and going out of the competition, they're going to throw men forward so if we had got that two-goal cushion we would be sat here through to the next round, but because we didn't and because we slipped up a little defensively, it's a replay.
"When you get so close and get to injury time, all you have to do is make sure you keep your composure, I think we lost a little bit of it just at that final minute.
"There was a bit of luck in it getting through so many bodies to Lukaku, but we should have done better before that and made sure we didn't get into that position."
Allardyce was pleased with his team’s display up until that point though, building on their recent good form in the Premier League and hoping to kick-start a run in The FA Cup.
"It was a very good performance," added the former Bolton Wanderers chief.
"Away from home, and when it's The FA Cup, you know that Everton are going to throw caution to the wind, because if they don't do anything from an attacking point of view they're out of the competition.
"So it's a great shame after such a good performance and being 1-0 up and we'll have to try and finish the job at Upton Park now, but we're still in the hat."
