The Hammers competed well in the first half before the unfortunate Demel put the ball into his own net three minutes before the interval.
Mamadou Sakho quickly made it 2-0 early in the second period and, although West Ham reduced their arrears through a Martin Sktrel own goal, Liverpool hit back to run out convincing winners.
Luis Suarez scored a third for the Reds and the scoring was completed by another own goal, this time from Joey O'Brien.
Hammers manager Allardyce felt the scoreline was a harsh reflection on his side, saying: "I think we showed in the first half that we didn't deserve to be 1-0 down.
"The game at this level is 90-95 minutes long not 45. For me we didn't recover from that (second) goal going into the start of the second half and that was the disappointing thing.
"We could have taken a lot of encouragement from the first half and the chance from Modibo Maiga that was well saved (by Simon Mignolet) is one you want to go in when you come to the big boys, but I couldn't criticise as it was a super save off a very good header.
"The second goal killed us for a while, got Liverpool right up for it."
Allardyce was unhappy his captain Kevin Nolan was sent off for raking his studs down the calf of Jordan Henderson late on, yet Jon Flanagan escaped for a tackle on Stewart Downing.
"Kevin Nolan shouldn't have lost his cool but obviously he did and Stewart Downing is in hospital having his Achilles and calf stitched up from a nasty challenge from Flanagan," he said.