The 20-year-old was dismissed in the 23rd minute for an innocuous-looking tackle on Luis Suarez in the centre of the pitch.
On a hot day the Toffees worked valiantly to reduce the impact of their numerical disadvantage but their resolve was eventually broken by goals from Andy Carroll and Suarez in the final 19 minutes.
"I thought it ruined the game," said Moyes of the red card.
"There are a lot of questions that people ask about derbies, tackles and sendings-off but that wasn't down to a bad tackle.
"I would have been disappointed if it had been a free-kick and if he had given a yellow card you would have said 'what is that for?'.
"It would have been hard playing Liverpool 11 v 11 - it was a really tough game for us and it became tougher."
Moyes has history with Atkinson as he was fined £8,000 by the Football Association 12 months ago for confronting the official at the end of their 3-3 draw with Manchester United at Goodison Park.
"We've not had this referee since myself and Steve Round were fined one year ago against Manchester United," said the Scot.
"It was interesting that the PGMO (Premier Game Match Officials Ltd) sent him into both clubs this week to do the referee's appraisal prior to this game.
"It needs people who watch and play the game and understand it and write about it to see it (Atkinson's decision) because it is easy for me to say it.
"Too often people talk about the players not doing it right but it wasn't the players.
"I don't think anyone in this world thought it was [a sending off] but it is one of these things that we just have to take that it was wrong and move on. I don't know if we can appeal the red card."
Moyes admitted, however, that other factors contributed to their defeat.
"The team did everything but at the end of the day it wasn't the referee that lost us the game, we made a couple of defensive lapses," he added.