Shin's men took full advantage - Kim Young-gwon scoring from close range in the second minute of stoppage time, before Son rounded things off with Manuel Neuer caught up the pitch.
Germany for the first time since 1938, not out of the group.
Germany's failure to progress beyond the group stage mirrored that of three of the previous four world champions: Spain in 2014, Italy in 2010, and France in 2002.
"But we now are out and, of course, it's a case of absolute sadness and disappointment, but I think we have young players who are very talented and who have the opportunity to go forward".
Meanwhile, Germany's exit from the World Cup had Twitter buzzing.
"We were rightly eliminated from the tournament".
South Korea, for its part, needed a miracle to qualify for the next round; in the end, although neither side qualified, it was a match that will be remembered for years to come in the Asian nation.
"We will have to have discussions tomorrow, we'll see how it goes".
Germany's team of superstars could only look on in shock in the final minutes of the match as USA referee Mark Geiger called for the video assistant referee to intervene after the unmarked Kim had bundled his shot past Germany 'keeper Manuel Neuer from a corner early in stoppage time.
While their loss to Mexico could be explained by a lack of cohesion between defence and midfield when coming up against counter-attacks, there is little excuse for failing to even score against a Korea side who came into the game without a point.
"Since 2006, we have been among the best four teams, but today we have not put in a performance we are normally able to do".
"Our team in this match was missing the ease of play and the classiness that we normally display", Löw said.
The Asians, however, were also unsafe on the counter-attack and a couple of times carved opportunities to keep Germany goalkeeper Manuel Neuer on his toes.
With no option other than a victory to advance, Low inserted Mario Gomez and Muller for Khedira and Goretzka.
Germany lost on Wednesday despite dominating in possession - 69 percent to 31 percent - and leading South Korea in shots on goal.
From 30 yards out, it should have been fodder for World Cup-winning Neuer, but he fumbled Jung's dipping drive Neuer and scrambled desperately to clear.
Loew, who recently signed a contract extension, said he would take time to consider his future.