With Moyes not enjoying an entirely smooth start to his reign, Martinez's side head down the East Lancs Road three places above their north-west rivals.
And Moyes knows what a positive impression Martinez has made since his own move from Wigan.
"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."
Moyes confirmed striker Robin van Persie "has a chance" of being fit after sitting out three games with a combination of groin and toe problems.
"He is doing okay," said the United chief.
As Moyes knows to his cost, Everton have not won at Old Trafford for over two decades, a record he will attempt to stop Martinez ending on Wednesday night.
"Old Trafford has always been a difficult place for anyone to play," he said.
"That is why we have been disappointed with some of our own results there.
"No-one sees Manchester United away as an easy fixture."
Despite his outstanding efforts on behalf of the Blues, Moyes' time at the club is not entirely remembered with affection by some Evertonian fans.
They were unhappy with the manner of his departure and following that were less than impressed when Moyes attempted to sign Leighton Baines and Marouane Fellaini, a pursuit that wasn't completed until the latter man changed clubs for £27.5million on deadline day.
"My contract was running out and I only found out about the Manchester United job four weeks before the end of the season," said Moyes.
"If you said how you would leave a club, I don't know if I could have done it any better. I don't know if anybody could.
"At the first right opportunity I tried to speak to Bill Kenwright and that is how we handled it.
Yet he accepts the reception he receives from the visiting fans may not be entirely positive.
"It could be mixed," he said. "They know their football and understand exactly where their club has gone.
"More importantly, they understand the players who are in their team. I brought the majority in and they will understand there are a lot of good players. But it is football and you never know."
Fellaini has yet to impress his new club's supporters and is still to score, despite hitting the net on a regular basis at Everton.
But Moyes is not concerned, adding: "It is the same settling-in process that most Manchester United players go through.
"I don't think it is just Marouane or David Moyes. It is players joining this club.
"Obviously there will be players who hit the ground running and can go right in but there will be others who take a little bit of time.
"There have been quite a few of those here."
Moyes can certainly do without dropping more points after successive draws at Cardiff and Tottenham.
Yet, for all the scrutiny on him, the Scot would not wish to be anywhere else.
"I knew the enormity of the job at Manchester United. It hasn't changed for me at all," he said.
"It is the biggest job and for me personally it is a great challenge."