Ultimately penalties were needed to decide the outcome of the quarter-final and it was Germany that triumphed 6-5 in the shootout after seven players had missed from the spot.
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Germany started the match in control of the possession, but it was Italy that had the first two attempts inside five minutes - Stefano Sturaro firing wide from a half-cleared corner, before Emanuele Giaccherini had an effort deflected behind after pulling on to the isolated Jerome Boateng.
Alessandro Florenzi had the game's first shot on target in the 15th minute as Italy threatened on the counter-attack once again, but the wing-back's effort was comfortable for German stopper Manuel Neuer.
Germany continued to pass the ball in the Italian half of the field, but they were forced into a 16th-minute change as Sami Khedira picked up a thigh problem in a challenge with Giorgio Chiellini, which allowed Bastian Schweinsteiger to enter.
Germany's first sight of Gianluigi Buffon's goal arrived in the 20th minute when Mario Gomez met a sweeping pass from Toni Kroos, but the striker's tame effort looped into the arms of the waiting Italian goalkeeper.
Joachim Low's side had the ball in the back of the net in the 27th minute when Schweinsteiger headed a Thomas Muller cross past Buffon, but the referee disallowed the effort for a push on Mattia de Sciglio.
Buffon mis-handled a cross from the left as Germany had another half-chance 10 minutes before the end of the first period, but the Juventus goalkeeper just managed to sort his hands out ahead of the waiting Gomez.
Gomez again had a sight of goal in the 41st minute after meeting a cross from Joshua Kimmich, but the experienced striker missed the post with a glancing header, before Muller fired straight at Buffon from close range.
Italy had two chances of their own in the final moments of the first period, but there were no takers on Giaccherini's low cross from the left, before Sturaro had a powerful strike blocked wide of Neuer's post.
Muller had a wonderful chance to send Germany ahead in the 53rd minute after Gomez set up the Bayern Munich attacker inside the Italian box, but Florenzi produced a block to divert the ball wide of the post.
The pressure continued to rise as the hour mark approached and Sturaro, De Sciglio and Parolo all picked up yellow cards in quick succession as Italy struggled to deal with the rise in tempo from the world champions.
Italy had a half-chance in the 62nd minute when Parolo met a bouncing ball outside of the German box, but his effort was always moving wide of the post as the quarter-final remained all square in Bordeaux.
Germany made the breakthrough in the 65th minute, however, as Gomez found Hector with a low pass and the left-back's cross was deflected in to Ozil, who fired home for his 20th international goal.
Low's side should have scored a second three minutes later when Ozil found Gomez inside the box, but the striker could not beat the brilliant Buffon with a close-range effort as Italy somehow stayed in the quarter-final.
Graziano Pelle had a golden chance to equalise in the 73rd minute after a De Sciglio cross from the left, but the Southampton striker missed the target after nipping ahead of the German defence.
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Italy were level in the 78th minute, however, when Bonucci converted from the penalty spot after Boateng had handled inside the German box.
The Azzurri might have scored a second three minutes later when Eder set up Pelle just outside the German box, but the striker's effort was deflected wide of the post during a period of end-to-end football.
De Sciglio, who ran a marathon for Italy in a wide position, had the final opportunity of normal time, but his effort from distance hit the side-netting as the third quarter-final of Euro 2016 headed for extra time.
Germany were the more forward-thinking team in the first period of extra time, but Italy's defensive unit kept the likes of Muller and Julian Draxler at bay as the score remained 1-1 after 105 minutes of football.
Draxler had the best chance of the second half of extra time when he took advantage of a loose header from Andrea Barzagli, but the attacker hooked over the crossbar from close range, before missing the chance to set up Muller following a fast counter-attack from the Germans.
Lorenzo Insigne came closest for Italy in extra-time after spinning inside the German box, but his effort was straight at Neuer and the referee blew the final whistle with penalties needed to decide the quarter-final.
Incredibly, Simeone Zaza, Muller, Ozil, Pelle, Bonucci and Schweinsteiger all missed in the shootout, but the vital miss came from Matteo Darmian at 5-5 and it was Jonas Hector that stepped forward to win it for the world champions and set up a semi-final against either France or Iceland next Thursday night.