"I am completely satisfied," said Bayern coach Juergen Klinsmann. "Our commitment was total. We want to go past Hoffenheim or whoever else is there as quickly as possible."
Dortmund led after only two minutes on a goal by Nelson Valdez, but Brazilian Ze Roberto midway through the first half.
Bayern created numerous chances in both halves but failed to score until Klose put the game away with a pair of clinical finishes.
Hertha Berlin missed a chance to tie Hoffenheim on points when it drew at Arminia Bielefeld, 1-1.