The Scot was very happy with his side's performance under tough conditions and heaped praise on the goalscoring exploits of his players, who put five past the League Two outfit
The Merseyside club put their League Two opponents to the sword and, despite a penalty which threatened a shock comeback, were able to see the game out comfortably.
After the match, Moyes revealed he was happy to see five different players on the scoresheet and, despite admitting the pitch was a tough obstacle for his men, he was delighted with the overall result.
"It was a tough game for us at the start, but that’s what you expect from FA Cup ties, they’re never quite the way you want things to go," he told reporters.
"The pitch was quite difficult for the players to play on but overall it was a job well done.
"Cheltenham always had a chance and even when we were at 3-1 they had a chance and at 3-2 the game is a lot different.
"But to be fair we scored some great goals tonight, Ossie [Leon Osman] scores a great goal and Seamus [Coleman] does as well so we played well enough to win the game."