The loss was Germany's second Nations League match under Klopp, who took charge after the World Cup, and the new Bundestrainer was unhappy with both the result and the way Greece caused his side problems.
“Losing this game is not allowed to happen,” Klopp said afterwards.
“The result was not cool, I really didn't like conceding the goal. The problem is that we didn't react well to long balls. Felix can of course win the duel, the goal is unlucky. They basically only played long.”
Klopp also rejected the idea that Germany could simply rely on individual moments to solve the problems exposed against Greece.
“Football needs training, football needs patterns of play,” he said. “We can't just win the balls where we want to win them and then hope that someone gets out of the way.”
The defeat came during an unusual period for Klopp, who has assembled a 44-man squad across the current international window in an effort to gather as much information as possible about his options.
The Germany coach defended that approach despite the disappointing performance.
“The plan in these two and a half weeks is to collect as much information as possible,” Klopp said.
“Just imagine if we had brought exactly the same team here as at the World Cup. I don't know whether the game would have looked much different today. From my point of view, there was no alternative to doing it this way.”
Klopp will now have another opportunity to assess his wider squad in the upcoming matches against Serbia on October 1 and Greece on October 4.
Only Finn Dahmen, Philipp Treu, Nico Schlotterbeck, Yannik Engelhardt, Bambasé Conté and Serge Gnabry remain with the group from the first selection.
For Klopp, the Greece defeat was therefore both a first setback and another piece of information in the process of building his Germany side. The next two matches will give the second group of players their opportunity to provide an answer.