The Toffees will take on Arsenal at Goodison Park in the Premier League on Tuesday evening.
Koeman wants the fans to give their full support to the players and cheer them on to all three points.
Everton will head into the match against Arsenal on the back of a 3-2 defeat to Watford away from home at Vicarage Road in the Premier League.
The Merseyside outfit are not in good form at the moment and will find it hard to even pick up a point against Arsenal.
The Toffees are currently ninth in the standings with 20 points from 15 matches.
Meanwhile, Arsenal find themselves second in the Premier League table at the moment with 34 points from 15 matches.
The Gunners are just three points behind leaders Chelsea and will head into the game against Everton on the back of a 3-1 victory over Stoke City at the Emirates Stadium in the league.