Germany missed three penalties before triumphing 6-5 in a rollercoaster 18-penalty shootout as Hector slotted home his kick under the diving Gianluigi Buffon and raced into his teammates' arms.
They will face the winners of Sunday's match between host nation France and surprise package Iceland in a semi-final on Thursday.
“It's hard to put it into words, but I am overjoyed that it went in,” Hector said.
“There weren't many people left. I knew I had to take one at some point and I had my heart in my mouth.”
Germany also shook off the hoodoo against old foes Italy, winning for the first time in nine attempts.
“Si, si, si, the curse against Italy is broken,” Bild newspaper said.
The Germans maintained a remarkable record of shootout successes — you have to go back to 1976 for the last time they lost on penalties in a major tournament.
Joachim Loew's side had appeared to be heading for victory after Mesut Ozil lashed into the net to give them a 65th-minute lead, but in a moment of madness Jerome Boateng handled in the area.
Leonardo Bonucci calmly converted the penalty 12 minutes from the end of normal time and it ended 1-1 on 90 minutes.
With no goals in a turgid extra-time, Ozil, Thomas Mueller and Bastian Schweinsteiger all failed from the spot for Germany, who are usually so reliable from 12 yards.
But after Simone Zaza and Graziano Pelle missed for Italy and Bonucci was denied by Germany 'keeper Manuel Neuer, Matteo Darmian's failure to convert the Azzurri's ninth kick paved the way for Hector to become Germany's hero.
Baby-faced Joshua Kimmich, 21, also converted when bigger names missed.
“The experienced players missed and the younger guys came through, so that's quite positive,” Loew said.
Italy's campaign had shown promise with wins against Belgium and over reigning champions Spain in the last 16, but like so many sides before them they were undone on penalties by Germany.
Coach Antonio Conte said: “I'm very proud of what the team achieved here.
“The only regret is the penalty kicks. But for commitment to the shirt, I can't fault them.”
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 on the bench but was introduced after quarter of an hour to replace the injured Sami Khedira.
Schweinsteiger found the net just before the half-hour mark but his effort was disallowed.
Loew had sprung a surprise by bulking up his defence, dropping the previously excellent Julian Draxler.
Germany dominated possession but 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 low drive was deflected behind for a corner.
The opener arrived on 65 minutes as Gomez pounced on some slack defending by Alessandro Florenzi before finding Hector in space.
The full-back's attempted cutback came off Bonucci but fell to Ozil, who fired past Buffon from close range.
Germany could have gone further ahead when Ozil found Gomez in space with a great chip, but the striker's outrageous backheel 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.
It was the first goal Neuer had conceded at the tournament after four successive clean sheets.
But he would play his part in the shootout, saving from Bonucci and Darmian.