Everton manager David Moyes claimed the Merseyside derby was "ruined" by referee Martin Atkinson's decision to send off Jack Rodwell early in Liverpool's 2-0 win.
Atkinson, who had visited both clubs this week ahead of taking charge of the 216th meeting between the Merseyside rivals, dismissed Rodwell in the 23rd minute for what looked like a legitimate challenge on Luis Suarez.
Suarez went on to add Liverpool's second after Andy Carroll's first Premier League goal of the season put Kenny Dalglish's side on their way to victory after 70 minutes.
A visibly disappointed Moyes said: "I just thought the sending off ruined the game. You get lots of questions from people asking about derbies, tackles and sendings off but that wasn't even a bad tackle.
"I don't even know if we can appeal. I would have been disappointed had it been a free-kick and if he had got a yellow card I think people would have asked 'what's that for?'"
Moyes revealed he had not spoken to referee Atkinson, who felt the full fury of Everton's fans inside Goodison Park at half-time and at the final whistle, adding: "We've not had that referee since myself and Steve Round were fined one year ago against Manchester United."
On his side's showing, Moyes said: "The team did everything they could and I'm disappointed. It wasn't the referee that lost us that game because we had a couple of defensive lapses but it would have been hard at 11 against 11.
"In the end it became tougher and the players kept at it but at 11 against 11 it wasn't a bad game and we were having our fair share. The game just stopped at the moment of the sending off.
"There is often talk about the players not doing it right but it wasn't the players today. Jack is obviously upset. He is a young boy making his way in the game and it was a big day for him."
Liverpool boss Kenny Dalglish was pleased with the composure his side showed, even after Dirk Kuyt saw a penalty saved by Tim Howard late in the first half.
He said: "It was pretty even at 11 against 11 but it is hypothetical to ask what would have happened if there had not been a sending off because we will never know.
"I thought we were composed and thorough in what we tried to do. We knew it was going to be difficult and Everton put a real shift in. David might be unhappy with the result but I think he will be happy with the efforts of his players and we are delighted with the way we went about our work."