It was the first ever victory for the Greeks in 11 matches against the Germans, and they now lead their group on six points from two matches. Germany have one point.
The hosts were in control for most of the game on Sunday (Monday AEST) and had nearly 80 per cent possession in the first half alone, but were ultimately punished for sloppy defending, especially in the second half, which gave the Greeks space to attack.
Klopp, who took over in July after Germany's early World Cup exit, has now failed to win either of his two matches in charge following a 1-1 draw against the Dutch in Amsterdam on Thursday.
"I could have lived with us not winning, but you cannot lose such a game," Klopp said. "We have been together for a week now, all mixed together, and that needs time.
"The result is not cool, but it is absolutely part of the process.''
Germany next play Serbia on Thursday, before travelling to Greece three days later.
Despite a dominant first half, it was only after they upped their tempo and started pressing higher that Germany carved out scoring chances.
Felix Nmecha fired wide in the 54th, a minute after Greece keeper Konstantinos Tzolakis had palmed a Joshua Kimmich free kick over the bar.
The Greeks, however, almost scored in the 66th after Kimmich lost possession in his own half, but substitute Fotis Ioannidis fired just wide.
They did it better in the 74th minute with Kourbelis's deflected low drive that gave them the win to make it two victories in a row after their last-gasp 2-1 win over Serbia on Thursday.
In the same group, Netherlands earned a 2-1 victory over Serbia thanks to Mexx Meerdink's double strike on his second appearance for his country.
It was the first win under new coach Xavi Hernandez following last week's draw with Germany.
"It's a boyhood dream. I actually don't have any words for it," Meerdink said. "Everything is happening so quickly. You just don't realise what can happen in the space of a week.
"Sometimes you need a bit of luck. This time it went my way. Then you have to be there and be important for the team."
Joao Felix scored for the second consecutive Nations League match as defending champions Portugal edged past hosts Norway 2-1 to top Group A4 with two wins from two matches.
The Norwegians cancelled out Felix's 17th-minute lead with Erling Haaland's equaliser, before Goncalo Ramos's chip won it for the visitors.
Ben Davies scored a 53rd-minute own goal and Gustav Isaksen added another on the hour as Denmark eased past Wales 2-0 to earn their first points in Group A4.
In a politically charged encounter in Group B3, Ireland scored twice in three minutes early in the first half to ease past Israel 3-0.
Austria beat Kosovo 3-1 in the other game earlier on Sunday.
In other results, Lithuania and Azerbaijan drew 1-1 in Group D2, and Gibraltar and Andorra fought out a scoreless draw.
