The last hope of Germany in a repeat scenario of the previous game against Sweden was defeated in injury time - Yeon Gwon Kim led South Korean forward, Heung-Min scored into an empty net.
Korea Republic will end a remarkable German run if they defeat the defending champions as the four-time winners have emerged victors in each of the five matches they have played Asian opposition at the FIFA World Cup.
"No matter what we say now, it will not be right", Muller told reporters. We just did not use our chances.
In the day's other games, Brazil went through at the top of Group E after a 2-0 win over Serbia, while Switzerland will qualify alongside them by virtue of a 2-2 draw with Costa Rica.
Germany, meanwhile, will be desperate to show they have finally arrived at the tournament after two indifferent performances against Mexico, which they lost 1-0, and Sweden.
Wrote one, "Korea just got Mexico to the next round of the World Cup". Originally called out for offside, the goal was given after video review.
Fellow supporter Denis, who travelled nearly 3,000km (1,864 miles) from his home in Stuttgart to watch Germany take on South Korea in Kazan, was equally resigned.
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The defeat also marked the first time Low has failed to reach the semi-finals of a tournament during his 12-year spell in charge.
"From the 65th minute we became incredibly hectic, played bad passes and were hit by many counter-attacks".
Marco Reus, who hit a crucial leveller in the 2-1 win over Sweden, saw his drive deflected and Mesut Ozil saw his effort deflected off a Korean boot and out for a corner, after which goalkeeper Hyeonwoo Jo smothered desperately when Goretzka's header found Hummels trying to pounce on the loose ball.
In a must-win match for the reigning champions, Germany could not live up to the expectations as a resilient South Korea team managed to oust the champions with a 2-0 win. "How we go from here is something we have to talk about", he said.
And Neuer, who only returned from injury in time to captain Germany at this World Cup, insists Die Mannschaft can not dwell on past achievements.
Loew was appointed national team coach after the 2006 World Cup in Germany, when he was assistant to Juergen Klinsmann.
"We didn't deserve to win the title again, we didn't deserve to make the last 16".