Robert Lewandowski's 72nd-minute equaliser against his former club and a penalty five minutes from time from Arjen Robben maintained Bayern's unbeaten record in the league this season with a 2-1 triumph at the Allianz Arena on Saturday.
Dortmund led in the 31st minute through Marco Reus - a seemingly inevitable contribution from the Germany international after Bayern's reported interest in his services dominated the pre-match build-up.
But Klopp is now left to reflect on a return of one point from Dortmund's last seven league matches - a run that leaves them third bottom of the table.
"It was close. It was a good first half and a less good second, in which we buckled to the pressure," he told his post-match media conference.
"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."
Midfielder Henrikh Mkhitaryan was central to much of Dortmund's impressive work before half-time and the Armenian felt he and his team-mates went on the defensive too readily during the second period.
"Bayern had a lot of chances," Mkhitaryan told reporters.
"We dropped too deep in the second half and that was the reason we lost."
