Referee Mark Clattenburg sent off both Tony Hibbert and Phil Neville for conceding second period penalties when first Hibbert brought down Steven Gerrard, and then in the final seconds, when Neville handled a shot from Brazilian midfielder Lucas on the line.
And, after the whistle, Everton players surrounded referee Clattenburg as he left the pitch.
Kuyt could easily have received a red rather than a yellow card but the Holland international remained on the pitch and was the difference between the two sides as he scored twice from the penalty spot.
In the first period Everton had dominated and were unfortunate to only have a Sami Hyypia own goal to their name at the break.
Moyes was furious afterwards and said: "I am not sure the first incident was a penalty. It was a coming together of two players and Gerrard's arm goes across Hibbert before there is any contact.
"The referee pulled out a yellow card and then the Liverpool captain has a word with him and it becomes a red.
"Decisions are made, they happen, that is football. But in the last seconds of the game there is the chance for it to be corrected and it would have been a result Everton deserved.
"We deserved that penalty if not more, and if the other penalties were more blatant than that, then I am in the wrong game.
"I am seemingly seeing football differently. If the referee does not see that you have got to ask why. Maybe they should not be there."
He added: "We train every day, we know what decisions should be and the referee does not give it. That is beyond everything I can imagine.
"Referees do their job as best they can, I have tried very hard in the past year not to talk about referees. We have had referees into training and we are very supportive.
"They have a tough job, but do they understand how tough my job is and what it would have meant for us to get those points in the bag?
"There are now a couple of players banned. We understand what happened to Phil Neville, that was correct but we got no decisions at all.
"It can't just be me being bitter, people asked was the referee biased and I can't comment but we got nothing at all.
"Even down to nine men we still had the chance of a penalty and we did not get it. A win would have pushed us into the top half of the league because of crucial decisions going against us."
Liverpool boss Rafael Benitez, predictably, saw things differently. He said: "People are talking about a penalty they could have been given. But last season when we lost here 3-0, two of their goals were fouls and nobody complained then.
"I always feel that in England, players should not be rewarded with penalties for diving."
He added: "I felt both penalty decisions were right. And I also agree with the yellow card for Kuyt and not a red.
"He did not touch the Everton player and was moving across him rather than straight into him, he was trying to charge down the ball.
"I felt we deserved to win and the performance of the players was fantastic. This result could turn our season and send us into two very hard games this week against Besiktas and Arsenal full of confidence."