The Toffees went into the match on the back of a 3-0 home loss to Tottenham five days earlier and with Koeman stressing they would “need to react” in Italy.
But they were abject once again last night, with their sloppiness punished by a dynamic Atalanta side who scored three times in the first half through Andrea Masiello, captain Alejandro Gomez and Bryan Cristante.
“A lot went wrong, except maybe after a difficult start, maybe 10 minutes that we had a little bit more of the ball and looked a bit more comfortable,” Koeman said last night.
“Ball possession was poor, but most painful was in the first half that they showed more aggression, that they showed more passion and that’s really painful because in my opinion it starts with that and then you need qualities on the ball.
“It’s a good question to the manager, because I asked myself the question ‘what am I doing wrong?’ about the second half against Tottenham, and the first half, because it’s the way I like to see my team.
“We can play better football, missing chances to score, but the commitment, the passion that they showed to us in the first half was much bigger than we showed together.”
A trip to Manchester United on Sunday continues Everton’s tough start to the season, but Koeman added: “We knew from the beginning of July that the start of the season was really tough.
“OK, we need to accept it, but that’s not the problem. The problem is what we showed in the first half together.”
Everton produced another hugely disappointing display in Reggio Emilia.
While failing to mount anything really resembling a fightback after the break, Koeman’s men at least managed to avoid conceding any further goals, although Remo Freuler’s shot did connect with their crossbar.
Everton will have hoped the fact this match was taking place at Mapei Stadium in Reggio Emilia rather than at Atalanta’s Stadio Atleti Azzurri d’Italia home — which is to be redeveloped to make it fit for European competition — would mean last season’s Serie A surprise package were less potent than they would have otherwise been.
But it was clear from the off that the Bergamo outfit, roared on by their vociferous fans, were going to provide a considerable challenge.
The visitors — with six changes to their starting XI from the Tottenham game — did threaten the Atalanta goal in the 13th minute, with Gylfi Sigurdsson’s strike being parried by Etrit Berisha and Wayne Rooney then trying with a delicate shot from the edge of the box that drifted just off target.
But otherwise they were more or less entirely on the back foot in the opening stages — something they were handling well enough until the 27th minute, when Masiello broke the deadlock.
The defender had a shot turned behind in fine style by Maarten Stekelenburg but from the resulting corner, the ball ricocheted off Phil Jagielka to the back post where the unmarked Masiello tapped in with ease.
While Everton struggled to muster a response, Atalanta continued to go at them.
And after Stekelenburg had been called into action a couple more times, the impressive Gomez made it 2-0 in the 41st minute as, with Everton standing off him, he received the ball from Andrea Petagna just outside the box, cut inside and unleashed an excellent shot that flew into the top corner.
Three minutes later Atalanta made the most of more ponderous Everton play, extending their lead via Cristante’s finish, and Koeman and his men must have been simply relieved to then hear the whistle for half-time.
Rooney had a shot blocked at the start of the second half but it was Atalanta who continued to show the real verve in attack, with Petagna having a low drive batted away by Stekelenburg and Freuler hitting the bar.
Koeman subsequently brought on Davy Klaassen, Sandro Ramirez and Kevin Mirallas but it made little difference to a thoroughly dispiriting night for Everton and their fans.
Berisha, Toloi, Palomino, Masiello (Caldara 76), Hateboer, Cristante (Kurtic 82), Freuler, de Roon,Castagne, Petagna (Cornelius 79),Gomez.
Gollini, Orsolini, Haas, Bastoni.
Stekelenburg, Holgate, Jagielka, Keane, Baines, Besic, Schneiderlin (Sandro 67), Rooney (Klaassen 66), Vlasic, Sigurdsson, Calvert-Lewin (Mirallas 76). Subs Not Used: Pickford, Williams, Martina ,Gueye.
Vladislav Bezborodov (Russia).