The Hammers were leading 2-1 heading into the final minute, but just as they had in their opening-day defeat by Cardiff, they were defending deeper and deeper and were punished when Adam Clayton grabbed a dramatic equaliser.
"When you are 2-1 up at home you have to take responsibility for seeing the game out," said Hammers chief Allardyce.
"You don't put pressure on yourself and unfortunately we did. We don't do what needs to be done to get the result, we've lacked the professionalism.
"Away from home it's not been a problem, the opposition haven't had a sniff, but here it puts a fear factor in everybody before they turn up at the next home game.
"That fear factor means everyone at the stadium is going to be edgy next time and that makes it very difficult for us to win matches.
"To win two away and then falter at home is a great shame but that anxiety will not go away until we win a game."
West Ham took the lead from a corner in the sixth minute when Carlton Cole volleyed past Andy Lonergan.
Leeds won a penalty in the 28th minute when Matt Taylor controlled the ball with his arm in the box but Max Gradel, the subject of a failed bid from West Ham earlier in the summer, rolled his effort wide.
Leeds finally drew level in the 59th minute when Robert Snodgrass' cross found McCormack in front of goal, but three minutes later the hapless Patrick Kisnorbo thumped Julien Faubert's cross into his own net.
West Ham should have gone further ahead but Scott Parker, being watched by England boss Fabio Capello, hit a post.
And the inevitable happened as stoppage time approached. Snodgrass raided down the right again, Jonathan Howson's shot hit the underside of the crossbar and Clayton followed up to fire past Robert Green.
Allardyce was also left fuming at referee Michael Oliver, who waved away two penalty appeals, firstly when Cole was felled by Aidy White and then when Kisnorbo appeared to wrestle James Tomkins to the gound.
"Two major decisions have gone against us today," he added.
"In critical parts of the game we should have had two blatant penalties. I've seen them again with our technology and nobody will convince me they are anything but blatant penalties.
"It's disappointing when you see the referee not do what he is paid to do.
"There's no doubt he's looking at the incident and making a judgment very quickly, and unfortunately on two occasions he made the wrong one."
Leeds boss Simon Grayson knows just knows how Allardyce feels.
"It's not easy being the favourites for the division, everybody wants to beat you, we had it in League One," he said.
"You could sense that nervousness creeping in, but you have to deal with it.
"But full credit to my players, we dug deep and a draw was the very least we deserved."