The Toffees followed up their 3-0 loss to Atalanta in their Group E opener by picking up just a point at Goodison Park against Cypriot side Apollon, who had led before claiming a late leveller shortly after being reduced to 10 men.
Koeman blasted his side's nervy opening 20 minutes, during which time they fell behind for the seventh time in their last eight games to Adrian Sardinero's opener when Ashley Williams gave the ball away.
Wayne Rooney capitalised on a gift at the other end to level and impressive substitute Nikola Vlasic put the hosts ahead after the break, yet Hector Yuste claimed a point for the visitors two minutes from time, moments after Valentin Roberge had been dismissed for standing on Dominic Calvert-Lewin.
Ronald Koeman, Manager of Everton reacts during the UEFA Europa League group E match between Everton FC and Apollon Limassol at Goodison Park on September 28, 2017 in Liverpool, United Kingdom.
Image credit: Getty Images
Koeman's side, who entered the new season with so much optimism surrounding them, have won just twice in nine games and there plenty of groans around Goodison at the pedestrian pace they exhibited early on.
"They are afraid in (how they are) playing," Koeman said of his players.
"That's difficult in the situation like we are - we are missing seven, eight players (through injury). Maybe that's too much in the situation that the team is. We've had two wins and a win today makes a difference.
"It's in the heads of the players, you can help them, support them, make it clear for everybody, give the players confidence, but finally by the whistle I don't play the ball back. It's up to the player - give the player options.
Ronald Koeman, Manager of Everton reacts during the UEFA Europa League group E match between Everton FC and Apollon Limassol at Goodison Park on September 28, 2017 in Liverpool, United Kingdom.
Image credit: Getty Images
"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."
Everton are currently propping up Group E with one point, though Atalanta's draw with Lyon means no side has more than four after two games.
Koeman saw this as a missed opportunity ahead of a double-header with Lyon, who reached the last four of this competition last season.
"If you don't win at home, you need a win or at least to draw away," Koeman added.
Everton's Dutch manager Ronald Koeman looks on before the UEFA Europa League Group stage match between Everton and Apollon Limassol at Goodison Park, in Liverpool on September 28, 2017.
Image credit: Getty Images
"I'm disappointed because we had the three points in hand and we deserved the three points. It's a difficult situation."
Apollon once again proved to be no pushovers having held Lyon to a 1-1 draw in their first contest.
Their manager Sofronis Avgousti hailed Yuste after he headed home their late equaliser having handed Everton a way back into the contest with an inexplicable back-pass that went straight to Rooney to score.
"We came here and tried to play football - it was a difficult game but we did our best," Avgousti said.
"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."
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