"We had it in our own hands, but we did not make it", Neuer, who confirmed his intention to remain available for global selection, said.
"There are some things we need to correct, but I will not talk about it publicly". The team, littered with stars such as Toni Kroos and Mesut Ozil, began by losing to Mexico 1-0 in their first game and only followed that up with a late victor against Sweden in their second.
The four-time winners came to Russian Federation as one of the favourites but became the fourth side this century to lose in the group stage after winning the previous tournament following a 2-0 defeat against unfancied South Korea.
With Germany desperately looking for a goal to get back in the match, Neuer joined the attack.
Defending champions Germany have been eliminated from the 2018 World Cup after losing to South Korea in Kazan.
The dense apartment complexes that shape capital Seoul shook with roars an hour after midnight on Thursday as South Korea scored two stoppage time goals to stun Germany 2-0 in what became both teams' final match in Russian Federation.
But it was Kim Young-gwon who broke the deadlock in added time when he found himself alone in front of goal after a corner and he poked the ball home from close range.
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Iceland pushed forward, and threw men forward in search of a victor which looked increasingly like it would send them through. The match was evenly poised in the final 10 minutes of the first half, but chances were few and far between for both nations.
Low, however, was adamant the future of German football is still bright, especially going by their results since the 2006 World Cup and their current youth prospects.
The 58-year-old only recently signed a contract extension last month that would keep him as Germany manager until 2022, but following his side's group stage exit, he refused to confirm if he will remain in his post.
The first two games had brought narrow but disappointing defeats against Sweden and Mexico and criticism had been growing back in Seoul.
After defeats to Sweden and Mexico, South Korea were expected to bow out of the tournament against a Germany side who needed a win, but a major surprise occurred in Kazan.
"We had a good training camp", Low added.
"We've done a fantastic job today with this match", he said in Yekaterinburg, ahead of a last-16 clash against either Brazil, Switzerland or Serbia.
However, after two late goals from the South Korean side, the Germans will head home early.
And Neuer, who lost the ball in midfield for the second goal, has admitted the 2014 winners just weren't good enough in Russian Federation. Sweden and Germany both had three points, and South Korea had zero.