Spaniard Thiago Alcantara, only back this month from a one-year injury absence, scored the decisive spot-kick for the Bavarians, runaway leaders in the Bundesliga and through to the Champions League last eight.
Without injured wingers Franck Ribery and Arjen Robben along with Bastian Schweinsteiger and David Alaba, Pep Guardiola's team were at times on the backfoot and an aggressive Leverkusen threatened them repeatedly.
Bayern were also lucky not to have Thiago sent off in second half stoppage-time after his flying studs-up challenge on Stefan Kiessling who had to be taken off.
Mario Goetze missed Bayern's best chance when he was left unmarked at the far post but could not score from three metres out.
They stayed calm and converted every penalty with Josip Drmic having his spot-kick saved by Manuel Neuer before Thiago scored.
Earlier, third division Arminia Bielefeld stunned in-form Borussia Moenchengladbach, also on penalties, after a 1-1 draw at the end of extra time.
Bielefeld, who claimed their third Bundesliga scalp, matched Gladbach, third in the Bundesliga, throughout and had a chance to snatch victory late in extra time as they showed no sign of fading against their more illustrious rivals.
Gladbach's Ibrahima Traore missed the decisive penalty as Bielefeld beat another Bundesliga club after eliminating Hertha Berlin and Werder Bremen earlier in this competition.
Borussia Dortmund, last season's finalists, booked their spot with a 3-2 extra-time victory over Hoffenheim on Tuesday and VfL Wolfsburg edged past Freiburg 1-0.
(Reporting by Karolos Grohmann, editing by Ed Osmond)
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