The Gunners won 2-1 to keep up their title challenge, but Louis Saha had put the visitors ahead on 24 minutes when he slotted home - despite being offside.
However, the Gunners turned the game around through substitute Andrey Arshavin and Laurent Koscielny's header.
Moyes, though, was left less than impressed by the Arsenal skipper.
"I think it was offside, the wrong decision, but it was not Everton who made the decision," Moyes said.
"I also thought there were some really bad tackles.
"But Fabregas' comments to the officials when he was coming down the tunnel warranted a sending off.
"I am not going to repeat what they were.
"They were disappointing comments from someone who is such a talented footballer."
Moyes continued: "I won't go into what they were, but they were deserving of a sending off, 100%.
"If you had said it on the pitch, you should have been off like that, so what is the difference when you are coming down the tunnel?
"We heard it, but that should not be the talking point."
Saha was offside when Seamus Coleman chipped the ball through, but it spun off Koscielny's boot as he attempted to clear, falling to the Frenchman who slotted a low effort past Wojciech Szczesny.
Despite the protests of the Arsenal players, and consultation with his assistant Stephen Child on the far side, referee Mason allowed the goal to stand.
Moyes continued: "If the linesman does not give offside, we have to take the chance. It is up to the officials, but it seemed to change a lot of things in the game."
Everton had lost just once since November before this game.
And Moyes felt his men had done enough to get a positive result.
"I thought the players did a sterling job, but we gave it away by our defending at a set-piece," he said.
"We have got nobody else to blame but ourselves.
"The players did really well and except for a lapse at a set-piece, they did a really good job."
Arsenal manager Arsene Wenger, meanwhile, was perplexed by the whole scenario. He said: "For me the goal was offside. If someone can demonstrate to me that it was mathematically not, I will accept that.
"The pass was towards Saha, who was three yards offside. They interpreted Koscielny's touch as a pass, not a deflection.
"He finished fantastically well, so we were 1-0 down and had it all to do."
The Gunners boss, though, praised the spirit of his men to again dig deep and find a result.
Wenger continued: "Tonight was a victory of a team with fantastic spirit and a never say die attitude rather than our usual game, but that is needed in this championship.
"I am very happy because the spirit is maybe the aspect of the team that's been questioned the most, and that's where we have been the most convincing recently, in smaller (cup) games against Leeds, Ipswich and Huddersfield.
"We played our 10th game of the month tonight and have shown, where our football has dropped a bit, a fantastic attitude."
Arshavin had been criticised for some below-par performances in recent weeks, but showed great composure to take his goal with a close-range volley.
"If you look at the assists in the Premier League, Arshavin is the best," Wenger said.
"He had a drop in form recently, but he is a great player.
"You could see he can deliver something special straight away when he came on.
"He is coming back. He was better on Sunday (against Huddersfield) calmer, and got us a vital goal tonight. I'm very pleased."
Wenger confirmed midfielder Samir Nasri would be out for 21 days with a hamstring problem, so will miss the Champions League first leg against Barcelona.
Alex Song (leg) and Theo Walcott (ankle) will also be assessed ahead of Saturday's trip to Newcastle after picking up knocks.