The Hammers were denied a first Anfield triumph in 50 years when Lucas turned James Tomkins' goalbound header off the line five minutes from time.
Nevertheless, Allardyce's team are edging themselves away from trouble and, unlike Stoke and Norwich, seem to have played themselves into a little bit of form.
"I'd like another three points," said Allardyce. "It depends on the run of the teams below us because some are picking up two or three wins back to back. But we are looking up, not down."
Tomkins and fellow centre-half James Collins were outstanding, restricting Liverpool to few clear-cut chances, even if one of them was ruled out for offside, only for TV replays to suggest Daniel Sturridge's second-half effort should have stood.
"To have won would have been a massive bonus for us as it is 50 years since we last did it," said Allardyce.
"But it is still a very big point, and very well deserved. I don't think we have defended better away from home all season.
"Collins and Tomkins were really good when you consider how few chances Liverpool had."
Tomkins was not able to see out the entire game, although Allardyce revealed the calf injury that forced his early departure was not too serious.