Atalanta scored all of their goals before the break thanks to Andrea Masiello, Alejandro Gomez and Bryan Cristante, the trio putting things out of Everton's reach after 44 minutes.
It was Everton's third successive defeat across all competitions following losses to Chelsea and Tottenham in the Premier League, and alarm bells are starting to ring for Koeman, who spent heavily to bolster his squad during the transfer window.
"The team is far away from what we need to show," Koeman told reporters. "I think most of the players are not at the level we expect.
"It's now time that we showed this is really the wake-up call. It's a big lesson today.
"I thought it was happening [the wake-up call] after the third goal against Tottenham, but okay - I need to ask myself the question, what I did wrong.
"I don't see a team on the pitch. I see 11 individual players. We have some new players, players that need time to adapt. They are young.
"You can find many more excuses, but I don't like that. I like to be realistic."