Moyes fielded a strong side against the League Two outfit, and the visitors took the lead in the 12th minute when Nikica Jelavić followed up Marouane Fellaini’s shot which had rebounded off the post.
Leighton Baines added a second from the spot before half-time, with Leon Osman virtually ending the third-round tie as a contest when he fired a low finish past Scott Brown.
Cheltenham did nick a goal after a neat move dissected the Everton defence and Russell Penn beat Tim Howard at the goalkeeper’s near post.
But late strikes from Seamus Coleman and Fellaini wrapped up the victory and set up a fourth-round meeting away to Sunderland or Bolton Wanderers.
“The manager played his full-strength team,” Neville told BBC Sport.
“The manager wasn’t happy at half-time and we had to step it up. The early goals killed their enthusiasm but they never gave up.
“David Moyes told us yesterday he wants to win the FA Cup. We’ve got one game a week now all season so tiredness shouldn’t be an issue.”
The Everton manager also expressed his displeasure at a tough fixture against the Black Cats or the Trotters in the next round, with the pair set to face each other in a replay at the Stadium of Light next week.
“It was a tough game for us which is what you expect in the FA Cup,” he added.
“The pitch was difficult but we got the job done. They always had a chance, but we scored some good goals tonight. We played well enough to win the game.
“I’ve seen it again and I think it is a penalty. The boy climbs on Fellaini’s back and you can’t do that. Other people can disagree but I would have given that.
“We’ve not scored a lot of goals this season so that helps us. If you don’t get great draws it’s not easy to win the FA Cup and whoever we play it’s a tough game for us.”
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