Moyes was delighted to sign off for the season with a win at St James' Park thanks to Charly Alcarez's second-half winner.
Moyes told evertontv: “I think the result is maybe as good as we've had (since I returned), because of the position we were in. We couldn't really move up or down in the league table today.
“I think it said more about the players who are playing for Everton today, in terms of what they put in and the character they've got.
“Since I've come in, I can only tell you that the players have been brilliant – their effort, their commitment, their drive.
“Today, we tried to work out how we could make it difficult for Newcastle, because they're coming to qualify for the Champions League today but I thought the players’ mentality was so good.
“If they keep that up, then it will give us every chance. I've said many times we might lack a bit of quality, but it's not the heart, it's not the commitment.
“Some of the players today were immense, they really were. There's quite a few players there who may not be here next season and they didn't flinch once, which is a credit to their professionalism.
“I think that’s a little bit of what this Club does to you. You can't give in, you will not give up on it. It's not accepted.”
I wasn't sure we'd stay up
Moyes admits he's full of pride for turning the team's fortunes around since succeeding Sean Dyche.
He continued: “I do (take pride from it), because I genuinely wasn't sure if we could do that when I came in.
“So what I want them to do now is I want us to take this absolutely into the new stadium.
“I want us to bring good atmosphere in. I mean I have to say the atmosphere inside St James' Park today was incredible, it really was. I hope that it's us who in the future, we're the ones who are challenging for Champions League and we can show them exactly what we've got.
“The away support was brilliant again and we’re thankful we've been able to give them something. We’ve won a few away games and they can travel home today and have a few drinks and enjoy it.
“The support has been magnificent, I've got to say, whether it be at Goodison or away from home, I think it's been great. So now we need to all come together and make sure this is going to happen at the new stadium because we're going to need it.”