The world champions will meet France or Iceland in the semi-final in Marseille on Thursday, after winning their sixth consecutive shoot-out at a major tournament following a 1-1 draw after 90 and 120 minutes.
Germany had taken the lead just before the midway point of the second half, as Mesut Ozil netted his 20th international goal from a Jonas Hector cutback.
Neither side looked like netting a winner in extra-time, so to penalties it went.
Neuer stopped two penalties in the shoot-out, from Bonucci and Matteo Darmian, and Graziano Pelle and Simone Zaza also missed as tired minds affected even more tired legs.
But Die Mannschaft’s reputation as spot-kick kings also took a beating, with Thomas Muller, Ozil and Bastian Schweinsteiger all missing from 12 yards.
Before tonight, Germany last missed a penalty in a shoot-out in 1982, when Uli Stielike failed to score from the spot in the World Cup semi-final against France.
In sudden death the quality improved, but when Darmian had his shot saved by Neuer, Hector kept his cool to slot home the winning kick and clinch it 6-5.