Everton twice went in front through Leighton Baines and Victor Anichebe, but substitute Demba Ba staked his claim for a regular place in the team by rescuing Newcastle with two goals, including a last-ditch equaliser to steal an unlikely point.
The Toffees will undoubtedly feel hard done by after referee Mike Jones disallowed two goals from Marouane Fellaini and Phil Jagielka which should have been given.
"There are two goals missed and we can't get them back. I had a word with the officials but I was not in there ranting and raving," Moyes told BBC Sport.
"I said 'you have missed two goals' but what can I do about it? We played very well and should have won the game deservedly but didn't.
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"Even from where I was I thought Marouane Fellaini's goal was a goal so I was surprised when it was given offside.
"We should have been out of sight by half-time and when it was still 1-0 Alan Pardew was probably delighted."
Moyes accepted it was a difficult decision to make to disallow the goal after Jagielka's header looked to have crossed the line, but was frustrated that the linesman was unable to see from a good position.
He told Sky Sports: "It's a difficult call but if you are the linesman, then you are on the line and you would hope they would see the ball go over the line.
"That is their job. He's looking right along [the goal line] so you would hope he would see it."
Moyes was also disappointed that his team allowed the lead to slip twice, especially after scoring what looked like a late winner.
"Yes we do [feel aggrieved]," he added.
"But we've got ourselves to blame for that and for different reasons, not just us conceding a late goal but we thought we had other goals in the game which should have counted and didn't on the night."
Nevertheless, the former Preston boss was satisfied with the way his team played, and conceded that Newcastle were the better team in the second half.
He continued: "The team played great all night. We got a goal then gave the ball away badly to give them a goal back to make it 1-1.
"Newcastle were the better side in the second half, but we scored a couple of goals which should have counted and didn't. Sometimes you need decisions to go your way but they didn't tonight.
"They weren't good [defensively] and we know we can do better. But I think there was an awful lot of positives in our play tonight.
"The defending at times could have been better but you have also got to give Newcastle credit. They put us under pressure and forced us into mistakes."