The Hammers were beaten 4-1 in their Premier League match at Anfield, but Allardyce has claimed that his side were causing more of an attacking threat before the Reds took the lead.
"There was an element of luck that got Liverpool on their way," he told the club's official website. "They weren't looking like they were going to score before half-time and, if anybody had the better chances before they scored, it was probably us.
"Unfortunately their keeper has made a couple of good saves and there were a couple of good blocks, then there was a ball through from Mohamed Diame to Modibo Maiga which just the slightest touch [may have put it in the net] - he just cannot seem to pull it off at the moment, the lad.
"Then he had a chance off this head which was a great header across the defender and their keeper has pulled off a wonderful save. To top it off, they took the lead from an own goal after Jussi [Jaaskelainen] had saved it when there was no Liverpool player there to tap it in."
West Ham United are currently 17th in the Premier League table.