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."
Hoffenheim are a point ahead of Bayern after their 1-1 draw at Borussia Moenchengladbach on Saturday. Hertha Berlin are third with 37 and Hamburg fourth on 36.
Dortmund struck after only two minutes when Nelson Valdez picked up a well-timed Tamas Hajnal pass, swept past slipping Bayern defender Martin Demichelis and curled the ball from the edge of the box past keeper Michael Rensing.
Bayern quickly recovered, unleashing a string of attacks at a shaky Dortmund defence with Klose often too quick for his marker Neven Subotic.
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.
Weidenfeller did well again to save a close-range strike from Ribery on 24 minutes but Ze Roberto was at hand to fire the rebound home from four metres.
CLINICAL FINISH
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."
In the other match of the day, Energie Cottbus climbed to 15th with a convincing 3-1 win over Hanover, who dropped to 13th with two goals coming after Hanover were reduced to 10 men with Leon Andreasen booked twice for fouls on his debut.
