The referee failed to send off Reds midfielder Lucas Leiva and Carragher has accused him of bottling the decision.
In a typically feisty affair it was a busy afternoon for the referee at Goodison Park once again, with him having to make a number of decisions throughout the game.
One particular incident saw Liverpool midfielder Lucas Leiva escape a red card when he tripped Ross Barkley while on a yellow card, with Atkinson deciding to give the Brazilian a warning rather than sending him off.
And Carragher openly criticised him for that poor decision.
“He definitely bottled it,” Carragher told Sky Sports.
“Sometimes you can book someone early on in the first half and maybe you’re not quite aware. He was well aware, he’s just frozen in that moment.”
Past experiences have affected Atkinson decisions
The former defender has also suggested that the referee’s prior experience of Merseyside derbies may have affected his thinking.
Sunday’s game was Atkinson’s fifth in charge of this fixture, having taken charge of last season’s 1-1 draw at Anfield and Liverpool’s 4-0 win in the past as well.
He came in for criticism for a performance in the same game back in 2011, when he sent off Jack Rodwell for a challenge on Luis Suarez, with replays showing that it was in fact a perfectly excecuted tackle from the then Everton midfielder.
And Carragher says that history may have affected his judgement on Sunday.
“He’s refereed derby games before and sent off three in five games now,” he added.
“Going into the game he’s probably thought he was determined not to send anyone off. Early on he let a few yellow cards go, but he’s seen that and he knew, Everton and Liverpool players knew and I think he bottled it.”