Bayern, bidding to become the first team to win three successive editions of the competition, which was incepted in 1935, fielded a formidable starting XI at home to Eintracht Braunschweig.
David Alaba's free-kick on the stroke of half-time broke the deadlock for Pep Guardiola's men, before Mario Gotze completed a comfortable 2-0 victory
Sebastian Schuppan doubled the hosts' lead following a corner, shortly into the second half, and although Clemens Fritz's pile driver halved the deficit, Junglas' second goal of the game made it 3-1 after 83 minutes.
Bremen's misery was compounded when Fritz was sent off for a second bookable offence late on.
It was Bielefeld's second successive elimination of Bundesliga opposition, after they sunk Hertha Berlin in the previous round.
Monchengladbach had no such trouble against fourth-flight adversaries, though it took an inexplicable handball from Kickers Offenbach's Markus Muller inside his own box to enable them to go ahead.
Munich, Gladbach and Wolfsburg will join fellow top-tier teams Bayer Leverkusen, Borussia Dortmund, Freiburg and Hoffenheim in Sunday's quarter-finals draw.
