The four-time champions finished bottom of Group F after stoppage-time goals from Kim Young-gwon and Son Heung-min secured a famous victory for the Koreans in a dramatic conclusion in Kazan.
It is the first time the Germans have gone out of the finals in the group stage and left head coach Low fielding questions over his future.
“That’s hard to say. Of course, it takes a couple of hours to look into that,” said Low about his own position. “I’m shocked, I’m shocked because we didn’t manage to pull it off today.
“Where we go from here, we’ll have to talk about it calmly. It will take me some hours now to come to terms with it. I’m incredibly disappointed by this elimination.”
It is the fourth time in the last five World Cups that the defending champions have been eliminated in the group stage. France in 2002, Italy in 2010 and Spain in 2014 all suffered the same fate as the Germans who went into the tournament with the nucleus of the side that triumphed in Brazil four years ago.
In the other Group F match, in Ekaterinburg, Sweden clinched a 3-0 victory over Mexico, meaning a win against South Korea would have been enough for Germany to progress.
However, Die Mannschaft were once again far from at their best, with defender Mats Hummels particularly wasteful in front of goal.
Kim was not so profligate when he fired home from close range – although he was initially given offside before a VAR review sparked renewed celebrations – and Son completed the job.
Asked about Kim’s strike, which was allowed after Toni Kroos was adjudged to have played the ball backwards, Low said: “The player was in an offside position – that’s what I saw. But if the video referees have taken a different decision, then they will probably be right.”
Low and his players will return to Germany on Thursday, but with the coach dismissing suggestions that dark times could lie ahead. It is the first time in six tournaments in charge that the coach has failed to guide Germany to the semi-finals.
“Until quite recently, we have probably been the most consistent, in terms of performance, team in the past ten, 12 years,” Low said. “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.”
For opposite number Shin Tae-yong, there were mixed feelings with the capture of a scalp only partial compensation for his side’s exit. Shin, who had given South Korea a one per cent chance of victory before the game, said: “I feel great, but at the same time, I feel a little bit empty, so I feel a little ambivalent.”