Dortmund looked on course to end a four-game Bundesliga losing streak at the Allianz Arena after Marco Reus fired the visitors into a first-half lead over the champions.
"It was close. It was a good first half and a less good second, in which we buckled to the pressure," manager Klopp told reporters.
"We deserved something for the first half but not for the second. We've invested a lot but got nothing in return, and that's what we have to worry about.
"But many congratulations to Bayern on a deserved win."
Klopp's side stay 16th in the German top flight, having not won in seven outings.