Klose scored from close range after a bad clearance from Dortmund's Felipe Santana in the 87th minute and tapped in a low Franck Ribery cross in stoppage time, helping the champions go second with 38 points.
"I am completely satisfied. Our commitment was total," said Bayern coach Jurgen Klinsmann. "We want to go past (league leaders) Hoffenheim or whoever else is there as quickly as possible."
Klose came agonisingly close in the 17th when he headed Ribery's cross from six metres only to see keeper Roman Weidenfeller make a spectacular save.
In a scrappy second half, Bayern created several chances but neither Ribery nor Luca Toni managed to beat Weidenfeller, Borussia's best player.
The champions needed the clinical finishing of Klose in the final minutes to break down Dortmund's resistance.
The German international pounced on Santana's bad header, slotting home three minutes from time and was again well placed inside the box to score from Ribery's cross from the left.
"I am very angry," said Dortmund coach Juergen Klopp. "I didn't think it was necessary for us to lose here today. We should have got a point."
