A 56th-minute header from James Collins gave West Ham the lead in the FA Cup third round tie at Goodison Park on Tuesday.
That looked to have been enough to see the London side through to a meeting with Doncaster Rovers or Bristol City in the next round.
However, Romelu Lukuaku fired home an equaliser in the 91st minute, leaving Allardyce to ponder over what might have been.
"We could have done an awful lot better. We got a bit desperate and instead of defending properly we let Bryan Oviedo in and he is very lucky with the cross," Allardyce told BT Sport.
"It ends up with Romelu Lukaku, but we could have done better and defended like we've been defending.
"It is very disappointing when it is so late in the game. Had Enner Valencia put a free header in the net it would have clinched it and you would expect a player of his quality to do that.
"It is never an easy place to come but we were more than a match for Everton.
"We've had our pocket picked. 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.
"We're still in the hat for the fourth round and now we will have to finish the job at home."
The sides will meet again at Upton Park on January 13.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.