Austria won the section ahead of France but the Netherlands were already guaranteed to qualify for the last 16, doing so as one of the four best third-place sides after crumbling in Berlin.
The Dutch beat Poland in their opening fixture before a goalless draw against tournament favourites France, but underwhelmed in the German capital against Ralf Rangnick's well-drilled outfit.
"The France match went reasonably well, taking into account the outcome, we played rather well as a team (but) today the performance was appalling," Koeman told reporters.
"People played poorly in certain aspects... we have to track our opponents and that's not what happened."
Koeman said his team made a catalogue of errors against Austria, who took an early lead through Donyell Malen's own goal.
Cody Gakpo equalised and after Romano Schmid struck for Austria, Memphis Depay pulled the Dutch back level again, but Marcel Sabitzer then lashed home the winner in the 80th minute.
"I can list quite a few (mistakes)," continued Koeman.
"We started very badly in many aspects. We did not defend well. There were many openings for the opponent.
"We were not aggressive. We lost the ball as well, particularly at the start. We were really very bad."
Netherlands won the competition for their first and only time 36 years ago to the day in Germany at Euro 1988, but Koeman said his team had to improve to have another chance of glory.
"We need to find a response, we have to prepare for the next match," added Koeman.
"That could be the final knockout, we don't want that to be the case, but we will have to play better than we performed today (to avoid that)."
Koeman was visibly angry with the team's performance in the first half and substituted midfielder Joey Veerman after 35 minutes to try and shake up his team.
The PSV Eindhoven player, 25, appeared on the verge of tears as he watched the rest of the match from the bench.
"I had to intervene as coach after the first half an hour," explained Koeman.
"We gave away the ball a lot, and so did Joey, but I could have taken off more players."
Koeman noted the first half display by the whole side was poor as Austria dominated the ball.
"We didn't have a lot of possession so it was very difficult, there was no rest in the game," he added.
"We were running like headless chickens. It was awful, very bad, we did not control the match."
