Moyes took over from Alex Ferguson in July after an 11-year stint at Everton, during which he established the Merseysiders as a top-half Premier League club.
The Scot endured a tough start to life at United, winning just three of his first eight Premier League matches in charge.
But a 12-match unbeaten run in all competitions, which has stretched more than two months, has eased the pressure on Moyes, who has seen his side rise to eighth in the table.
Any finish outside the top four would be unacceptable for reigning champions United, but they are now just three points off local rivals Manchester City, who sit third.
They came from behind twice to claim a 2-2 draw at Tottenham on Sunday, with striker Wayne Rooney scoring both of United's goals.
Rooney has netted 10 times in 16 appearances for his club this season and is in excellent form - something that will delight Moyes given the absence of Robin van Persie, who is battling toe and groin injuries.
Moyes is expecting a win against Everton and knows they need to act quickly to keep the pace with Premier League leaders Arsenal, who lead United by nine points.
"We have to win all of our games. I'm concerned we're not as close as we'd like to be (to Arsenal) but it's a long season and hopefully we'll be there or thereabouts," he told MUTV.
"There are a lot of games to go. I think the Premier League is tight and we'll try to stay on the shirt-tails and put ourselves in a good position at the start of next year."
Midfielder Michael Carrick (calf) is sidelined for United, while Everton will be without Leighton Baines (toe), Arouna Kone and Darron Gibson (both knee).
The absence of Baines - a close-season transfer target of United - will be a blow, but Everton are in good form and sit fifth on the back of a six-match unbeaten streak.
They thrashed Stoke City 4-0 on Saturday and goalkeeper Tim Howard has kept four clean sheets in his last five matches.
Chelsea loanee Romelu Lukaku has scored eight in his 10 appearances since arriving at Goodison Park, and his season-long signing has been hugely important for Roberto Martinez's side.
Everton have a decent record against United in recent years, having won two of the last seven meetings and drawing another two.
United have not lost at home to Everton in their last 20 top-flight meetings, though, and won the corresponding fixture 2-0 last season thanks to goals from Ryan Giggs and Van Persie.
