The Premier League's third-placed side arrived in West Yorkshire as one of the favourites to lift the trophy, but left a rain-sodden Elland Road with a 2-1 third-round defeat after having their noses bloodied by the Championship battlers.
Moyes made six alterations to the team which won convincingly at Swansea on Saturday, giving breathers to the likes of Leighton Baines, Tim Howard and Phil Jagielka, but his starting line-up still featured the likes of Sylvain Distin, Marouane Fellaini and Steven Naismith.
But the Toffees, be it before or after Nikica Jelavic, Phil Neville and Steven Pienaar were thrown on as part of an unsuccessful rescue mission, were never able to settle on the ball as Aidy White gave Leeds a fourth-minute lead and the excellent Rodolph Austin doubled it with 20 minutes left.
Distin would reduce the deficit with a header but Everton had left themselves with too big a mountain to climb, something Moyes was quick to acknowledge.
"We were not very good in the first half, but we got better in the second, but not as good as we've been playing," he said.
"We started terribly, gave them the initiative and lost the game in the opening 20 minutes due to the way we started. We gave away a really poor first goal and left ourselves trying to get back into the game."
On whether or not his decision to rest some players was key, he added: "These boys are in the squad and have to be able to show they can come in and play. If you don't use them, what's the point in having them? The players want to play and be involved."
Moyes' assistant Steve Round spoke on Monday of the club's desire to eventually lift this competition which, despite their Premier League form, is widely viewed as their best chance of silverware.
That has now passed for another season at least, leaving Moyes to rue a missed opportunity.
"It's a chance if you get good draws and you play well," he said.
"If you get difficult draws it makes it harder and so does not playing well. We had a combination of that."
While Moyes headed back to Merseyside to plan for a pick-me-up meeting with Southampton, opposite number Neil Warnock showered in what he called his "best-ever cup win".
Such praise was rich from a man who took an unfancied Sheffield United side to the semi-finals of both domestic competitions a decade ago, but he claims that owing to the injury crisis that is currently sweeping his already-thin squad, this victory was sweeter.
"I think that's as good a performance as I've ever had in the cup, because of of the circumstances," he said.
"They had a strong team, they're nearly top of the Premier League and they wanted to win. But we were better on the night. We deserved the win. They did have a few chances and we did too.
"It was like going back in time tonight, with the weather chucking it down, the fans singing, the officials were great, everything was."
Like Moyes, Warnock also tinkered with his line-up, throwing in transfer-listed Danny Pugh and the sparsely-used Jamie Ashdown and Michael Brown on a night when he needed all his players to be at the top of their game.
"It had everything, Warnock added.
"We had our centre-halves in bandage and covered in blood and we had the likes of Browny, Pugh and Ashdown. They've hardly played but they fitted in lovely. I'm pleased for them all. There was no weakness because if there had have been, we wouldn't have won."