Moyes was in charge at Goodison Park for 11 years before joining United in July to take over from the retired Alex Ferguson.
He has not enjoyed a successful start at Old Trafford, however, and United are languishing seventh in the table, nine points and two places below Everton with only a slim chance of finishing in the top four.
"First of all, the game is not about me, it's about Everton and Manchester United," Moyes told a news conference on Friday.
"I go there as manager of Manchester United and I'm delighted to be that. I was proud to be manager of Everton at that time but my job now is to make sure I get a result for United.
"The Everton fans were always terrific in my time there - very supportive and always right behind the team. We go there with a great away record and go there trying to keep that going."
United have the best away record in the Premier League but their home form has been poor and Everton are one of six teams to come away from Old Trafford with three points this season.
"I know the players at Everton - they are a great group of lads and great professionals who got on with the job and were self-motivated," Moyes said.
"Over the years, we continued to add players there and it was a great group of young players. The senior group there is very important too.
"(Manager) Roberto Martinez has done a brilliant job this year. He's got them playing well and in a good, healthy league."
Moyes acknowledged that United's chances of finishing in the top four and qualifying for the Champions League are fading, although they have beaten Aston Villa 4-1 and Newcastle United 4-0 in their last two league games.
"We've said we'll never give in and we won't until we can't do that," he said. "Obviously, it's nearly mathematically impossible but we're still trying to make it happen if we can.
"It's a tough game going to Goodison for any team at any time, as I should know, but we go there with a good away record and will try to make that work again."
United striker Wayne Rooney is available after shrugging off the toe injury which hampered him in last week's Champions League second-leg defeat at Bayern Munich and former Everton midfielder Marouane Fellaini is also fit to return.
"Rooney trained great this week and has been in really good form," Moyes said. "We're hopefully getting him back as he wasn't quite right with his toe at times."
Brazilian right back Rafael and Dutch striker Robin van Persie will miss out again due to injuries.
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