World Cup holders Germany won the shoot-out 6-5 after a tense match finished 1-1 after 90 minutes and extra time, and their reward is a semi-final with the winner of Sunday’s last quarter-final between hosts France and Iceland.
Joachim Loew’s side had appeared to be heading for the win in regulation time after Mesut Ozil pounced to give them a 65th-minute lead.
But Italy clawed themselves back into the game when Leonardo Bonucci leveled from the penalty spot 12 minutes from the end of normal time after a Jerome Boateng handball.
After no goals were scored in extra time, Ozil, Thomas Mueller and Bastian Schweinsteiger all failed from the spot for Germany, usually so reliable from 12 yards.
But after Simone Zaza and Graziano Pelle missed for Italy and Bonucci was denied by Manuel Neuer, Matteo Darmian’s failure to convert the Azzurri’s ninth kick allowed Hector to become Germany’s hero.
The game had been billed as the pick of the quarter-finals, but large spells of the game in Bordeaux were uneventful.
Germany talisman Schweinsteiger again started the game on the bench but was introduced after quarter of an hour to replace Sami Khedira, who came off with a groin injury.
Schweinsteiger found the net just before the half-hour mark but his effort was disallowed for a foul on Mattia De Sciglio.
With a semi-final against either hosts France or Iceland up for grabs, a cautious approach from both sides was to be expected.
Indeed, Germany coach Joachim Loew sprung a surprise by moving away from his usual 4-2-3-1 formation in favor of a 3-5-2 to replicate the Italian set-up, dropping the previously excellent Julian Draxler as a result.
The World Cup holders went on to dominate possession throughout but Gianluigi Buffon was untroubled by a Mario Gomez toe-poke while the veteran goalkeeper also easily saved from a Mueller shot late in the first half.
Italy responded in kind, Emanuele Giaccherini seeing his cutback go all the way to Stefano Sturaro, whose his low drive had power but was deflected out for a corner.
Germany moved up a gear immediately after the restart, Gomez smartly laying the ball off for Mueller to hit a left-foot drive that was turned behind by the backtracking Alessandro Florenzi at the far post.
The Germans were beginning to control proceedings and the opener arrived soon after. Gomez pounced on some slack defending by Florenzi before finding Hector in space to the left of the penalty box.
The full-back’s attempted cutback came off Bonucci but fell to Ozil, who fired past Buffon from close range to give Germany a 65th-minute lead.
Germany’s confidence was now soaring and Ozil found Gomez in space with a great chip, but the striker’s outrageous flick was superbly saved by Buffon and the offside flag came up anyway.
Italy looked down, but they were not out.
An unmarked Pelle got his left foot to Florenzi’s low cross but fired off target, and minutes later the Azzurri pulled level after the ball came off the arm of Boateng in the area.
Hungarian referee Victor Kassai immediately pointed to the spot and Bonucci stepped up to beat Neuer low to the goalkeeper’s left to spark wild celebrations.
It was the goal Neuer had conceded at the tournament after four successive clean sheets.
There was a minor scare for the Germans with three minutes left when Eder found De Sciglio on the left and the AC Milan midfielder saw his powerful drive ripple the side-netting.
Into extra time they went and Germany came closest in the additional half hour, substitute Draxler pouncing on a poor headed clearance from Andrea Barzagli to send an overhead kick just over, before both sides settled for penalties.