Liverpool boss Rafael Benitez and his Arsenal counterpart Arsene Wenger believed their Carling Cup game should not have been postponed.
Referee Martin Atkinson decided to call the quarter-final off due to heavy fog.
Benitez said: "I think we could play. "You cannot change the decision. You must respect it but we don't agree."
Wenger added: "It's disappointing because the young players had a good opportunity to play. I think it would have been very interesting."
With both teams involved in four competitions, no date has yet been given for when this Carling Cup tie can be played at Anfield.
Liverpool and Arsenal are also due to meet in the FA Cup third round.
Benitez added: "Myself and Arsene Wenger tried to convince the referee that we could play, but it was impossible.
"The referee said that he could not see the different colours and that it would be difficult for the linesmen.
"But you could see both goals clearly from the centre-circle and the referee said no. I'm surprised because we felt we could play in those circumstances.
"The question now is that, if we draw against Arsenal in the FA Cup third round, when can we play the game? Maybe we need to play both cup ties in the same game!
"If the FA Cup game is a draw, then they will maybe have to move the League Cup semi-final to fit this game in."
Wenger said he would have happily played the match on Wednesday but there was not enough time to put everything in place.
"It is impossible in 24 hours to rearrange a game," he added.
Referee Atkinson said he had endeavoured to get the game played but he had to consider not only his fellow officials and the players but also a full house at Anfield.
"We were informed at 1600 GMT this afternoon the conditions were quite bad and we got to the ground at 1650 GMT to have a look at it," he said.
"It is very misty and has been for the last three hours. We have been in the stands, looked round from different angles from a spectators' point of view and the weather is not getting better - you can't see some parts of the pitch."