"He is doing a very good job," said Moyes. "He has very good players there. I always told them they could play without a manager because they are very well organised.
"But Roberto is doing a really good job keeping it going.
"I had a great time at Everton, it was a great club for me at the time in my career when I left Preston North End. They were fabulous to me - they gave me a chance to succeed the best I could at the club.
"We had one or two early years at the start when it was a bit up and down but overall we had a pretty good level of consistency after that and tended to finish pretty well in the division.
"I don't think it's any advantage there's a new manager there and he's got his own players in as well.
"I do know a lot of them, I probably brought most of them to Goodison, if not them all, except for the recent ones. They're all really good players and I've got a lot to thank them for because they helped me to win games."
Moyes admits the manner of his departure was difficult for him and Everton.
"I was running out of contract. I only found out about this Manchester United job about four weeks before the end of the season. If someone said 'how would you leave a club?' I don't know if I could have done it better - I don't know if anyone could have done it any better. At the first right opportunity I spoke to Bill Kenwright and that's how we handled it."
And Moyes is not sure how he will be treated by the Everton supporters.
"I think it could be mixed at times but I think they know their football and they know where the club's gone," he said.
"I think more importantly they'll understand the players in the team, the majority I brought in and I think they'll understand there were a lot of good players brought in. But it's football and you just never know."
For breaking stories and all the great banter like us on Facebook: facebook.com/tribalfootball