insists Tottenham skipper Jamie Redknapp should have been sent off for the foul which ended Tim Cahill's involvement in the North Londoners' 1-0 win at Goodison Park.
The flash point of a spiteful match came after 60 minutes when Redknapp caught Cahill high on his left leg, a challenge that only resulted in a yellow card from referee Graham Poll.
But Cahill, after several minutes of treatment, was taken off and refused to accept Redknapp's apology or offered handshake.
Moyes admitted the midfielder may have serious knee ligament damage and has ruled him out of flying out to join up with the Australian squad for two World Cup qualifiers against the Soloman Islands.
Moyes said: "I do not believe that Jamie Redknapp is that sort of player, but I must admit I felt the challenge deserved a red card.
"I have looked at it again and I feel that most referees would have felt it was a red card, not a yellow one.
"The lads here have done really well until now, they have done terrifically well and they have finished this game knowing they should have won.
"We didn't, and that happens. There were a few decisions that did not go our way and yes, there was a bad tackle. On this occasion it was a tackle that was worthy of a red card.
"Tim will not be able to go with Australia, we will need to have a good look at his leg. His leg has got a real 'wobble' so we do not know whether it is his knee or shin."
Spurs boss Jacques Santini defended Redknapp, and also Jermaine Defoe, who was booked for a two-footed tackle on David Weir that also upset Moyes.
The French coach said: "I do not think that either of the bookings, for Redknapp and Defoe, deserved more than yellow cards. I do not feel that either player intended to hurt or injure their opponents."
The irony wasn't lost on Jacques Santini as he recalled the recent draw against Norwich City after Saturday's 1-0 win at Everton.
Noe Pamarot's first goal for the club at the start of the second half clinched the three points and lifted us back up to fourth in the Premiership on Saturday night.
"It was a tight game but ironically we had several chances in our game recently with Norwich and did not take any," smiled Jacques at Goodison Park. "This time we had two efforts on target and we won. That's football."
Everton had the best of the first half and were unlucky not to take the lead - Tim Cahill hit the post, Leon Osman forced a fine save from Paul Robinson and Kevin Kilbane fired just wide. They also had two penalty appeals waved away.
Pamarot's goal arrived after 53 minutes. Robbie Keane worked a short corner with Simon Davies and crossed for the French defender to power home a header from 16 yards.
Pamarot soon turned saviour at the other end as Osman struck the post and Kilbane was set to tap home the rebound only for the full-back to beat him to the ball.
And that was about it from the home side as the arrival of Duncan Ferguson led to a longer game and we coped comfortably defensively.
Jacques added: "For the first 35 minutes we didn't play well because Everton are a very strong team and had chances to score.
"My players have spirit, we kept trying to play and as the game progressed we could see the possibilities for ourselves.
"We have three or four key players injured and our team is tired physically and mentally but to get a result like this will lift everyone. It was a good win and a good performance. We are delighted to win but Everton could have won as well - it was that type of game."
"This game was very difficult because we were playing them at a good time and they have had some good results," he said.
"But after 25 good minutes my team believed in the possibility of winning and I am very happy with the result and for where this takes us in the league.
"I hope our fans are happy.
"We defended well with good positions on the pitch."
The game was marred by a bad tackle from Jamie Redknapp on Tim Cahill, but Santini did not see the incident.
"When we are on the bench it is very difficult to see the action," said Santini.
"I don't think Jamie Redknapp is a very strong player, it was a situation that just happened in the game. Again it is difficult to see the situation, but the referee saw it and he gave a yellow card."