"He was a very lucky man to stay on the pitch for his tackle," Moyes said.
Asked if it was a red card offence, Moyes added: "Definitely. It's over the top, down the back of the leg."
Liverpool manager Brendan Rodgers defended his striker - who scored one of Liverpool's goals, was instrumental in the other, and had one ruled out for offside in stoppage time - claiming the tackle was mistimed rather than malicious.
"I didn't think it was a red card," he said. "Sometimes you get that with a striker, when they are just a fraction late. They are trying to time it just as they pass it."
Suarez 'goal' should have stood - Rodgers
A linesman's flag incorrectly denied Suarez a last-minute winner as he prodded in from close-range after a Steven Gerrard free-kick.
Moyes, who saw his side come from 2-0 down thanks to goals from Leon Osman and Steven Naismith, admitted the Toffees were fortunate with that decision.
"It was a goal but I don't think it was a foul on Gerrard in the middle of the pitch," he said.
"That was the wrong decision but it was also the wrong decision not to give the goal."
Rodgers was frustrated by the decision but said he had been impressed by the courage shown by his young players.
"Obviously we are bitterly disappointed," Rodgers added. "It is an easy decision as well.
"I am not going to go on about officials because it has been happening to us all season.
"It is actually quite straightforward - it is a free-kick from deep, the ball is in the air for a long time and the official is looking across the line, so he is clearly onside.
"When [Sebastian Coates] heads it back, Luis is minimum level and it is a wonderful finish. He should have had his goal - it would have been a brilliant win for us.
"We are frustrated but I am are very proud of my young team in a cauldron of an atmosphere against a very experienced team."
Suarez was never far from the action and provided the game's enduring image when he celebrated Liverpool's opener by running towards Moyes and swallow diving on the touchline in response to
pre-match comments made by the Everton manager.
Moyes reacted angrily to the celebration, approaching Suarez who was quickly surrounded by his team-mates.
When asked for his reaction, the Scot said: "I thought it was great. I actually quite liked that and it's the sort of thing I'd have done if I'd have scored."
Rodgers described the celebration as nothing more than Scouse wit and, when asked for his view on diving, referred to the example of Everton captain Phil Neville who was booked in the first half for a poor attempt at a dive - for which he later apologised.
"Do you want me to talk about Phil Neville?" Rodgers added. "Don't start me."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.