Moyes, appointed as Toffees boss in 2002, is yet to oversee a win at United with the Merseysiders' most recent victory there coming in 1992.
However, they head into Sunday's Barclays Premier League contest having drawn 4-4 on their previous visit in April and then defeated the Red Devils 1-0 at Goodison Park in August, a game which opened what has been a fine 2012-13 campaign for Moyes' men.
Everton, fifth in the table, have lost only three matches in the league all season and their manager is in optimistic mood.
"Someone told me it was 1992 when we last won at Old Trafford, so that tells you how difficult the task is," Moyes said.
"But we have been quite good this year in that we have not lost many games and always been in there.
"We won earlier in the season against Manchester United and we drew there at the end of last season, so we will go there probably in a better frame of mind and with a little bit more confidence than at any other time since I arrived here.
"We will go there and try to get a positive result."
Moyes also hopes the fact United are playing Real Madrid away on Wednesday in the first leg of their Champions League last-16 tie might prove a factor Sunday, adding: "It will be a hard job for us, but with a bit of luck, maybe they might have Madrid on their mind for Wednesday night and rest a few players."