Germany have failed to win two matches in a four-team group stage for the first time in 32 years, since this format was introduced to the World Cup, and only for the fourth time overall. They could also have gone through with a draw, if the Sweden-Mexico game had also ended in a draw.
With Germany's loss, Sweden and Mexico advance from Group F. Sweden beat Mexico 3-0 in their match, which was played concurrently with the South Korean shocker.
South Korea scored both of their goals during stoppage time, the first of which was initially disallowed for offside but that decision was overturned after a video review.
South Korea also avoided another shameful record in their history book as the latest result didn't lead up to four consecutive defeats in global matches, which would have been the first time for the senior national football team.
"The disappointment of being eliminated is just huge, but we need to congratulate Mexico and Sweden who have made it out of the group".
Shock, scorn in Germany as World Cup champions suffer humiliating exit
Low's future has been called into question despite signing a new deal before the start of the World Cup . After 16 consecutive World Cups, world champions Germany are going home .
Fourteen teams have secured spots in the knockout stage at the World Cup, leaving just two more up for grabs today.
With that win they held their fate in their own hands, knowing victory over South Korea would guarantee them of a place in the last 16.
Kim Young-gwon broke the deadlock in added time when he found himself alone in front of goal after a corner and poked the ball home from close range. "We advanced to second phase, and thanks to Korea!" said Mexico City resident Edmundo Robert, who was at the game with his son and father-in-law.
Mats Hummels headed over the top when unmarked six yards out in the closing stages, before South Korea's two late goals.
Since making their World Cup debut in 1954, South Korea have only finished the tournament with three losses only once. Mexico next faces the victor of Group E and, Alcocer concedes, "We have to more careful about everything".
South Korea will be one of the favourites to win the tournament, which begins in early January and runs until early February - which is a blow for Tottenham Hotspur.