South Korea had the first chance when Jung Woo-young's free-kick was fumbled by Neuer, who punched clear bravely as Son tried to convert the rebound.
It is the first time since 1938 that Germany has not advanced out of the group stage of World Cup play.
However, if Germany and Sweden both win, that would result in a three-way tie on six points, which would be determined by the same measures described above.
Germany and South Korea remained knotted at 0-0, but one goal from the German side would spell ruin, and Toni Kroos had already proven his team was more than capable of a late go-ahead goal.
President warns Harley-Davidson: 'We won't forget'
Rising trade tensions and retaliatory tariffs could pose serious risks for Harley-Davidson, rating agency Moody's said on Tuesday. Trump then threatened the company "will be taxed like never before" if they follow through with the move .
Germany is out of the World Cup, the fourth defending champion in the last five tournaments to be eliminated during the group stages.
To advance to the round of 16, South Korea needed to beat Germany by at least two goals and have Mexico beat Sweden in the other Group F match.
South Korea was also eliminated despite the victory.
Should Neymar and Brazil stumbled in Moscow, Switzerland are hoping a win over Costa Rica in Nizhny Novgorod on Wednesday could catapault them into the last 16 as Group E winners. "The situation is very hard to put into words". It looked as if South Korea would be heading home with zero points and three defeats from three games.
22-year-old winger Julian Brandt, who was controversially included in the Germany squad ahead of Manchester City's Leroy Sane, may have earned himself a starting place ahead of Julian Draxler after impressing from the bench against Sweden.
England v Belgium. What will the score be?
He added he was concerned about the prowess of England captain Kane, who is the tournament's top scorer with five goals so far. Belgium will present England's biggest test yet at Russian Federation 2018. "That's what is brilliant about tournaments.
Their prayers were answered when late goals from Kim Young-gwon and Son Heung-min secured a stunning win for South Korea, and ensured Mexico and Sweden progressed to the knockout stage at the expense of the Germans.
Jerome Boateng, who was sent off against Sweden, will be replaced and many now anticipate that we will begin to see the Germany we have become accustomed to at major tournaments in this final group game. "Of course it's disappointing because we lost in the group but I think I can be proud after this win and we can be proud of this win".
"Maybe we need one more match and he will be in his full form physically and technically, because he is outside the normal standards". Suele made his first World Cup appearance as a replacement for Jerome Boateng.
Gomez had a decent header stopped by goalkeeper Jo Hyeon-woo in the 68th minute and could not properly connect with a low cross from Joshua Kimmich in the 72nd as Germany's hopes vanished.
The person filming the video, Patrick Infurna, says "Only the World Cup could do this".
Should England fear Belgium?
The striker may well be rested for the mouth-watering clash against England after suffering a knock against the Tunisians. Training is conducted in English under Spanish coach Roberto Martinez, who spent a decade managing in England .