VfL Wolfsburg led a group of Bundesliga clubs who were knocked out of the competition, suffering a humiliating 3-1 home defeat to second division Energie Cottbus.
Wolfsburg's defeat put their coach Steve McClaren's future in doubt as they have now gone eight games without a win.
Fellow Bundesliga club Cologne lost to second-tier Duisburg 2-1 while Eintracht Frankfurt were eliminated by Alemannia Aachen on penalties.
Bayern built up a quick two-goal cushion by the eighth minute with Andreas Ottl thundering in from 30 metres and Mario Gomez notching his 19th goal in the last 15 games in all competitions.
Stuttgart, who had lost 5-3 to Bayern at the weekend, battled back and levelled on the stroke of half-time with Russian Pavel Pogrebnyak, who had also cut the deficit in the 32nd minute.
Substitute Klose, in his second game back since a 10-week injury absence, struck in the 52nd minute, taking advantage of a defensive blunder by Matthieu Delpierre to slot home.
RED CARDS
Stuttgart looked doomed when Khalid Boulahrouz was sent off with a second booking and Christian Gentner saw his penalty saved by keeper Hans-Jorg Butt.
But captain Delpierre, who himself was sent off three minutes from time, stepped up to level with a well struck 77th minute header.
Bayern made the best of their extra players and with late goals from Thomas Muller, Klose and Franck Ribery made sure of their spot in the last eight.
"Stuttgart gave everything they had. It was an unbelievable game," said Bayern coach Louis van Gaal.
"It was great game for fans in the stadium... and now we can truly enjoy the holidays."
Wolfsburg, who have not won any of their last seven league matches and languish in 13th place, were 3-0 down at half-time thanks to a brace by striker Nils Petersen.
The defeat makes the future of coach McClaren beyond the four-week winter break even more uncertain. The former England boss joined at the start of the season as the first English coach in the Bundesliga.
Cologne, with Lukas Podolski back in the team, were also left playing catch-up when they fell a goal behind after three minutes against Duisburg.
Duisburg added another goal late in the second half and held on despite Cologne's Simon Terodde pulling one back six minutes from time, to seal a memorable Cup win that allowed them to advance into the last eight.
Frankfurt missed once from the spot as Aachen converted all five penalties after extra time had seen the two teams locked at 1-1.
