The reigning German champions were on the way to their Shinji Kagawa's opener around the hour mark.
But it was not to be as two late goals from Karim Haggui and Didier Ya Konan eventually helped Hannover to the all three points.
Speaking after the match, Klopp bemoaned the poor result although he praised his side's performance.
"It was clear to see that the team had big plans, but in the end we're left empty handed and we only have ourselves to blame for it," he was quoted as saying on the club's official website.
"The way we played wasn't bad, but the final result is. The loss really hurts us, but we can only learn from this defeat.
"We outplayed them in every area if you look at the statistics, yet we failed to beat them."
Hannover coach Mirko Slomka was pleased with the way his team fought back after a dominant display by Borussia.
"Borussia were very dominant during much of the game," he said. "We were unhappy to fall behind but it was deserved.
"But we didn't stop playing. This team always believes in itself and was rewarded for a great fight."
Dortmund have dropped to 11th spot in the standings following this weekend's results.