The Toffees - in their second Group E fixture, a fortnight on from the woeful 3-0 away loss to Atalanta - struggled to show any real dynamism early on against Cypriot outfit Apollon and fell behind in the 13th minute when Adrian Sardinero's goal punished an Ashley Williams mistake.
Subsequent strikes from Wayne Rooney in the 21st minute and substitute Nikola Vlasic in the 66th had
Everton looking set for victory, but then a chaotic conclusion to the game saw Hector Yuste head an 88th-minute equaliser moments after Valentin Roberge's sending off had reduced the visitors to 10 men.
Koeman, whose side have won only twice in their last nine matches, said afterwards: "They are afraid in (how they are) playing.
"It's in the heads of the players. You can help them, support them, but finally I don't play the ball back. It's up to the player.
"We are scared, we are afraid to play forward and that makes it difficult.
"Maybe sometimes there are not enough options to play forwards but even if there are no options, play the ball in the channel instead of back to the goalkeeper.
"We know what the fans like, that you go forward, and they don't like to see a team playing back, back. In the first 20 minutes every ball was back and then it's really difficult to create chances."
With one point on the board, Everton remain bottom of Group E. Atalanta top the pool with four points after their 1-1 draw at Lyon, who are on two like Apollon, and now have a double-header against Koeman's side.
Apollon's dramatic leveller completed quite a turnaround for Yuste as he had earlier teed up Rooney's goal with a misjudged back-pass.
Apollon boss Sofronis Avgousti said: "We came here and tried to play football. We were very unlucky with the Everton equaliser but we showed mental strength and spirit and reacted well.
"I was almost certain the same player (Yuste) could score from a set piece and that's what happened."