The World Cup holders triumphed 6-5 in the shootout following the initial 1-1 draw.
Manuel Neuer's save from Matteo Darmian enabled Jonas Hector to convert the winning kick to send Joachim Low's side through to the last four.
Arsenal’s Mezut Ozil had fired Germany into a 65th minute lead before Leonardo Bonucci equalised from the penalty spot following a handball from Jerome Boateng.
Can and Germany will face either France or Iceland in the semi-final on Thursday July 7.
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