Moyes had rested players for the Capital One Cup tie with Championship side Leeds earlier this season and seen the Toffees beaten 2-1 at Elland Road.
But he paid League Two side Cheltenham the compliment of playing most of his star performers in a 5-1 away success last night to clinch a fourth round visit to Sunderland or Bolton.
Goals from Nikica Jelavic, Leighton Baines, Leon Osman, Seamus Coleman and Marouane Fellaini secured victory.
"We did a very professional job. I wanted to try and progress in the cup if we can," Moyes said. "We've got a bit of momentum and we want to try and keep it going if we can.
"We are only in one cup competition. It's not as if we've got the Champions League or Europa League or anything else to deal with as well.
"With the side I put out, it could have proved to be folly if we had picked up injuries. Then it would have been made to look a bad decision.
"But I was disappointed when we lost our League Cup tie to Leeds United. I gave a few players a go that night and I didn't think it came up to scratch.
"I wasn't going to take a second chance, that's for sure. I just felt we needed the people who knew what was going on."
Everton effectively killed off the tie in the first 20 minutes via Jelavic's close range finish and Baines' spot-kick.
Moyes said: "It's always hard a game like this because the Premier League at times is all-consuming.
"But I thought the players did a good job and kept their focus right. Their application was spot on."
Moyes rubbished speculation that striker Victor Anichebe was set to leave Everton and join Reading.
He said: "There is no truth in that. We won't be selling anyone, if I have my way, that's for sure."