The Merseyside club's top-four bid suffered a big blow on Wednesday as Everton slumped to a 3-2 home defeat to Crystal Palace, ending a seven-match winning streak.
Defeat ensured Arsenal remained fourth and leaves Martinez's men with a crucial fixture against the Premier League champions on Sunday in their continued push for a UEFA Champions League berth.
Martinez – who called the game "probably the most important" of the season – paid credit to Moyes for the work he did at Everton before leaving in the close-season.
But he wants Everton fans to leave their praise for former manager Moyes until after the final whistle, calling on the club's supporters to repeat the intimidating atmosphere that helped them beat Arsenal 3-0 earlier this month.
"We all know how important the game is and how vital the three points are," Martinez said on Friday.
"All the nice welcoming is going to be after the final whistle. We have got a great understanding of who is coming back but we have to understand they represent the opposition.
"We will show the respect to David Moyes. His work has been terrific. As a former Everton manager, every Evertonian has got great memories of the job he did with the football club.
"(But) we need to make it as hostile as we can for the opposition."
Victory would give Everton their first league double over United in 44 years, but Martinez says another triumph against Moyes will not be easy.
"Facing the champions is always a reason to be excited," he added.
"The opportunity to do the double on United is something that is not easy to achieve. In football it doesn't get any bigger than achieving something that takes so many years (to do).
"You need to remember, this team won the championship last season. You don't become a bad team overnight.
"There is always a bit of a transitional period as you get a new management team in place. They are really strong and we expect a very tough game on Sunday.
"We know that we are going to face a very strong team, it is more about how good we can be. And that is where the focus is."
Martinez feels that the two-year contract extension signed by ex-United goalkeeper Tim Howard on Thursday was an "incredible moment" for Everton.
He also commented on the fitness of Phil Jagielka and Steven Pienaar, who are both recovering well from hamstring and knee injuries respectively.
Jagielka may be fit enough to make an appearance off the bench, but will not be risked from the start.
