Everton were soundly beaten by a side returning to European competition for the first time in 26 years as the home team's dominant first half set up a comfortable 3-0 victory.
The Toffees conceded all three goals in a ragged first 45 minutes and never threatened a comeback, instead limping to a third straight defeat in all competitions.
Koeman, who presided over an expensive influx of new faces over the close-season, shouldered the responsibility for the performance despite suggesting his team needed to display more fight.
"[Atalanta] showed more aggression, they showed more passion. And that is really painful, because in my opinion it starts with that and then you need quality on the ball," Koeman told BT Sport.
"I asked myself what I am doing wrong, about the second half against Tottenham [a 3-0 defeat last weekend] and the first half here. It is not the way I like to see my team. The commitment and the passion, what they showed in the first half was much bigger than what we showed.
"But once again it is not the time and the place to criticise the players. I need to criticise myself because the team was not prepared and that is a question for myself.
"The whole Atalanta team was much better prepared than us. Once again that is the manager."
Everton are now without a win in five matches across all competitions after a difficult start to the Premier League season.
Having already faced Manchester City, Chelsea and Tottenham Hotspur, their task does not get any easier on Sunday (Monday morning AEST) with a looming trip to take on Manchester United at Old Trafford.
Koeman says the difficult run of fixtures is no excuse for a slow start.
"We knew from the beginning of July that the start of the season was really tough. We need to accept it," he said.
"That is not the problem. The problem is what we showed in the first half."