The Hammers hope to claim a first Anfield win since September 1963 by finding the away goals that have been lacking so far this season - and they must do so without Andy Carroll, who is blocked from playing against his parent club.
"Most of the time Liverpool were so much better than anybody else, you wouldn't have expected it," said Allardyce of the half-century wait.
"Other people have won there - West Brom and that - so they have slipped up now and again.
"You just hope when you go it's your turn to catch a big boy who slips up on the day.
"On any given day you hit your best form, then anybody can win any football match.
"It happens less often at some places like Anfield than it does others."
The Hammers beat West Brom 3-1 last weekend, with two goals from Carroll, to all but secure their Premier League survival.
Allardyce believes his side are capable of winning at European hopefuls Liverpool, but knows if they are not at their best they will be punished.
"We need to be on top form and we need to take our chances when they arrive," he said. "If we don't life will be really tough at Anfield."