His side have an extremely slender chance of qualifying for next season's Champions League, while hosts Everton are Arsenal's main challengers for a top-four finish.
Although the three points are important to both teams, Moyes's appearance on Merseyside will dominate the headlines. With that in mind, Sports Mole has picked out five of United's most memorable away victories over the Toffees during the Premier League era.
1. Everton 0-1 Man United, October 1993
It was an encounter that was largely dominated by the home side, but they were unable to find a way past United goalkeeper Peter Schmeichel. The Dane produced a number of saves, which meant that Lee Sharpe's goal for Sir Alex Ferguson's side was a winning one. It was a fine strike by the England winger as well, as he lashed a volley beyond a static Neville Southall from 25 yards out.
2. Everton 2-3 Man United, September 1995
Sharpe was again in good form at Goodison Park, scoring United's first and second goals of the contest. However, both of those were cancelled out by Anders Limpar and Paul Rideout respectively. It appeared that the Everton duo had secured their side a point, but the match was won in the closing stages by Ryan Giggs.
© Getty Images
3. Everton 1-4 Man United, October 1998
On this occasion, United were rampant as Dwight Yorke put them in the lead early on, before Craig Short put through his own net. Duncan Ferguson rose highest before the break to reduce the arrears, but Andy Cole fired in via the post in the 59th minute to restore United's two-goal advantage. Then, five minutes later, the points were sealed as Jesper Blomqvist headed in his first goal since his switch from Parma earlier that year.
4. Everton 3-4 Man United, February 2004
Having scored on his debut against Southampton, Louis Saha went even better next time out at Everton, scoring twice either side of a strike from Ruud van Nistelrooy as United took a 3-0 advantage into the break. However, that was wiped after the restart as Everton hauled themselves level courtesy of goals from David Unsworth, Kevin Kilbane and United defender John O'Shea, who put through his own net. The Red Devils, though, were not done and in the last minute Cristiano Ronaldo crossed to the back post for Van Nistelrooy, who headed in from close range.
5. Everton 2-4 Man United, April 2007
It's the match viewed by many as the defining moment during the 2006-07 title race. While United's rivals for the trophy Chelsea were being held to a 2-2 home draw by Bolton Wanderers, Ferguson's men were overturning a two-goal deficit. Alan Stubbs and Manuel Fernandes scored either side of half time for Everton, only for O'Shea and an own goal from former United defender Phil Neville to draw the scores level. Then, with 11 minutes left to play, Wayne Rooney scored against his boyhood club, before youngster Chris Eagles made sure of the outcome in stoppage time.