The German champions fell to two quickfire goals from Mario Mandzukic and Thomas Muller, thus letting their five-match winning streak over the Bavarians slip.
However, while the Schwarzgelben created a number of clear-cut chances in a productive second half, they could only pull one goal back, through Robert Lewandowski.
"We went missing in the first half," Klopp explained to ZDF. "The result is okay, because we played much better in the second half.
"We had some big chances."
Meanwhile, Marco Reus, who was making his Borussia Dortmund debut following his summer move from Borussia Monchengladbach, was left frustrated by his new team's performance.
"We were not compact enough and could not put them under pressure," he said. "We have to improve even more because we cannot play like we did today."