The decisive goal came in the 74th minute. Substitute Dimitrios Kourbelis scored with a deflected shot after good work by Christos Tzolis on the right. The ball ricocheted off Joshua Kimmich before finding the net, giving Greece a famous victory.
Germany dominated possession for long periods at the WWK Arena but struggled to create clear chances. Karim Adeyemi had an early opportunity and Kimmich tested the goalkeeper from a free-kick, yet the hosts lacked a clinical edge.
Greece, newly promoted to League A, remained organised and dangerous on the counter. They had already shown their quality with a 2-1 win in Serbia earlier in the week and made it two victories from two matches.
Klopp, who had drawn 1-1 with the Netherlands in his first game in charge, made changes in the closing stages including the introduction of Serge Gnabry, but Germany could not find an equaliser.
The result leaves Germany with just one point from their opening two fixtures, while Greece sit top of the group with a maximum six points. Klopp’s side will look to respond when they face Serbia next before travelling to Greece later in the window.