Germany dominated possession and created numerous opportunities, but Greece defended resolutely to secure a point and remain ahead of the visitors in the group standings.
Greece make early threat
The hosts created the first clear opportunity of the match inside the opening minutes.
Georgios Masouras found space in the German penalty area, but goalkeeper Jonas Urbig produced an excellent save to keep Germany from falling behind.
Germany gradually took control of the contest and spent much of the first half camped in the Greek half.
Germany dominate but fail to score
Florian Wirtz, Derry Scherhant and David Raum were among the players who threatened as Germany looked for an opening.
Germany eventually found the net through Jonathan Burkardt in the 32nd minute, but the goal was ruled out after VAR identified a handball involving Tom Bischof earlier in the move.
The visitors continued to create chances and finished the first half with 15 attempts, but Greece goalkeeper Konstantinos Tzolakis stood firm to ensure the score remained 0-0.
Greece withstand second-half pressure
The tempo dropped after the interval as Greece became more compact and made it increasingly difficult for Germany to find a way through.
Philipp Treu had a good opportunity from close range but opted to square the ball rather than shoot, while Scherhant also came close as Germany continued searching for the breakthrough.
Klopp introduced Said El Mala and Nick Woltemade in an attempt to provide fresh attacking impetus, but Greece's defensive organisation remained intact.
Germany were ultimately unable to turn their territorial dominance into a winning goal.
Germany remain third in Group A2
The draw leaves Germany on five points after four Nations League matches, with Greece and the Netherlands both on eight points.
The result means Klopp's side still have work to do if they are to secure a place in the Nations League quarter-finals.
Germany's next Nations League fixtures come in November, when they face Serbia away before hosting the Netherlands in their final Group A2 match.
Match summary
• Greece: No goals
• Germany: No goals
Final score: Greece 0-0 Germany