The hosts missed a number of opportunities to secure the win before the shootout was needed, but Allardyce was quick to pay tribute to opposing goalkeeper Mark Howard, who made a string of impressive saves.
"The fans know we should have won the game," the Hammers boss told reporters. "We opened them up time and time again, but the end product deserted us. At the end of the day it is our fault.
"It was an inspired performance from their goalkeeper - whatever we threw at him he saved it or blocked it."
West Ham return to action against Southampton on Saturday afternoon.