The Hammers boss questioned referee Anthony Taylor's decision to award the league leaders a second penalty but did concede the hosts were lucky with their first half equalizer.
The league leaders left Upton Park with all three points after Steven Gerrard converted a penalty in each half to keep their title challenge on track.
Allardyce questioned referee Anthony Taylor's punishment of Adrian for a challenge on Jon Flanagan for the second penalty - with television replays confirming the West Ham goalkeeper had got a touch on the ball first.
The manager, though, conceded his side got some luck on Guy Demel's equalizer, when Liverpool goalkeeper Simon Mignolet dropped Mark Noble's corner under a robust challenge from Andy Carroll.
"Unfortunately we are talking about the officials and not the game," Allardyce told Sky Sports. "We took the game to Liverpool whenever we could and really tested them. There were fouls given by the referee when he was 60 yards away; the second penalty was not a penalty, Adrian plays the ball. Controversially, we are talking about the referee and the effect he had on the game.
"He made big mistakes in the game. He thinks Adrian hasn't played the ball and he has. He has to be 100 percent certain. He was in a bad position but gave it. It looked like Andy Carroll fouled the goalkeeper for our goal but we got it. We don't want that, nobody wants that. My real concern is that we are talking about referees every week. We don't have enough quality referees. They try their best but something has to be done about it."