Paraguay striker Roque Santa Cruz converted the decisive spot kick to give the home side a 5-4 win in the shootout, which settled a dull tie that was goalless after extra time.
Schalke's captain, Denmark striker Ebbe Sand, contributed to Bayern's qualification for the quarter-finals when his opening penalty bounced off the top of the bar.
Bundesliga leaders Bayern, reduced to chasing domestic glory after bowing out of Europe, took revenge for their semi-final defeat by the same opponents last season.
The first half was a tight, tactical affair with few highlights after Bayern goalkeeper Oliver Kahn brought down Belgium striker Emile Mpenza inside the penalty area a few seconds into the game but escaped unpunished.
Bayern stepped up a gear after the break but kept bumping into a compact defensive wall and hardly created a clear chance.
Substitute striker Elber came close to scoring with a superb volley from inside the box five minutes from the end of extra time in the most spectacular moment during a poor game.
Mpenza fired just over the bar from a counter-attack
Bundesliga strugglers Nuremberg, whose Macedonian striker Sasa Ciric missed a penalty just before the break, went out after a 2-0 home defeat by second division leaders Cologne, striker Matthias Scherz scoring both goals in the second half.
Four-times Cup winners Werder Bremen needed a strike one minute from time by substitute Ivan Klasnic to snatch a 2-1 win at second division LR Ahlen.
Greek striker Angelos Charisteas had put Werder ahead on 16 minutes, but the home side responded with a headed goal from midfielder Marc Arnold in the 74th minute.
Crisis-hit Kaiserslautern also went through with second-half goals from defender Hany Ramzy and midfielder Lincoln for a 2-0 home victory over second division SC Freiburg, who had knocked out German champions Borussia Dortmund in the previous round.
Bayer Leverkusen, last season's losing finalists, advanced on Tuesday after needing a penalty after the break for a 2-1 victory over second division Waldhhof Mannheim.
SpVgg Unterhaching, the only amateur side left in the competition, reached the last eight for the first time with a 3-2 win over first division Hansa Rostock, also on Tuesday.