James Garner, Hayden Hackney and Thierno Barry scored for the hosts, with Dan Burn grabbing a late consolation for Newcastle at Hill Dickinson Stadium.
Grealish was omitted from Everton's squad after the club confirmed he had breached its disciplinary rules, with the winger reportedly failing to attend a scheduled training session.
Moyes refused to discuss the details but insisted the decision had his full support.
"You probably know I am not going to give any details. It is a club decision and I support it."
Asked whether he felt let down by Grealish, the Everton manager added: "It is something I don't want to speak too much about."
Moyes was nevertheless pleased that the situation did not become a distraction for his remaining players, who produced a convincing performance to eliminate the Premier League opposition.
"The one thing about Everton is that they have got real good values, all the players stand behind it and accept the values, on the pitch and off the pitch. I think the players did a good job tonight, it is a small squad but we are doing our best to keep challenging.
"It was a tough game but I think the Premier League is like this, games are quite close, you can be defending for 20 minutes and then attacking. We got the goals at good times, the third goal killed the game off but really pleased to get through, you can talk about cup games, the only thing that really matters is you want to get through."
Garner opened the scoring in the 34th minute with a curling effort after Hackney helped the ball into his path following Tyrique George's pressure on Amar Dedic.
Hackney then doubled Everton's advantage shortly before half-time, converting a cross from Barry with a first-time finish.
Newcastle United manager Matthias Jaissle was frustrated by the manner of his side's defending, although he believed the final scoreline was harsh on his players.
"We are of course disappointed with the result. It's also a little bit shocking because, overall, the game was not that bad.
"But it is the story of the season so far with a situation where it really counts, we are lacking in the clinical approach to solve the situations.
"Just think about the first one, this is again a low-risk zone where we said, clear the ball, nothing happens, and a couple of seconds later you can see the goal out of nothing."
Jaissle was equally unhappy with the circumstances surrounding the second goal, believing Newcastle should have received a penalty before Everton punished them on the counter-attack.
"The second one is also frustrating, because in my point of view we deserve to have a penalty on the other hand, and then you miss that in the transition to defend their own box, which is again for me difficult to accept, we have clear rules on how to behave in these matters.
"And the third one, again a critical ball loss when you're in possession, and the position is punishing you on that level, pretty clear."
Barry completed the scoring in the 69th minute, finishing after Kiernan Dewsbury-Hall broke forward and delivered the ball across the penalty area. Burn headed Newcastle's consolation in the 88th minute.
Despite the defensive problems, Jaissle insisted his team had created enough opportunities to make the contest considerably closer.
"These are the things we need to learn quickly, and on the other hand I think you all saw also the game and the statistics.
"We had many, many occasions, many opportunities to score, big chances as well, and also in their box we need to be more clinical to get a different result."
Newcastle also suffered an injury concern involving William Osula, who was unable to continue after twisting his ankle while attempting a shot.
"William Osula was not able to continue, he twisted his angle in that occasion when he took the shot, and with the other two as well, we wanted to have a bit more fresh energy at the end," Jaissle said.
Moyes dismissed suggestions that the prospect of a quarter-final against League Two Fleetwood Town had provided extra motivation.
"I didn't phrase it that way and say anything to the players at all. This was a big Premier League game really and trying to win any isn't easy.
"What we have next you have to take in your stride. Anyone who beats Arsenal is doing a good job."
Everton will now prepare for another meeting with Newcastle in the Premier League on Monday, while the Carabao Cup victory has secured them a place in the last eight.