Hiroshi Kiyotake's curled free-kick just after the hour was enough the secure an unlikely win for the visitors, their first at Signal Iduna Park since 2008.
Klopp's men have now lost five of their last six games in the Bundesliga, but the coach remains defiant.
Speaking after the game, Klopp said: "We will continue to fight. There is no alternative.
"We had four or five great chances, but nothing made of it. We need to reward ourselves [with goals], but we're not doing that."
Dortmund's profligacy in front of goal was summed up by attacker Marco Reus, who had seven shots on goal, but still could not find a way past Hannover goalkeeper Ron-Robert Zieler.
The Germany international keeper was also kept busy by Mats Hummels first-half header, but he scrambled it onto the crossbar.
Six Dortmund defeats represents the most losses from any side in the league, and is in stark contrast to their UEFA Champions League form which has been the 2013 finalists win all three of their games - the most recent coming in the shape of a 4-0 thrashing of Galatasaray.
Klopp added: "We were in the decisive moments once again, but were not consistent enough. The popular explanations do not apply."