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Germany finished third in the standings on Friday (AEDT), behind Spain on goal difference, with Japan top following their 2-1 win over the Spaniards.
The result left Japan facing Croatia in the last 16 while Spain will face Morocco.
On a night of unrelenting drama in the desert near Doha, the Germans struck first with Serge Gnabry in the 10th minute but saw the Central Americans score twice with Yeltsin Tejeda and Juan Pablo Vargas, before a Kai Havertz brace put them back in front.
Fellow substitute Niclas Fullkrug added a fourth in stoppage time but it could do nothing to prevent their elimination as Spain advanced with a superior goal difference.
Germany and Spain both took early leads in results which would have put Germany through.
But it all changed in the second half with Japan scoring two quick goals against the Spanish, including a debatable second from Ao Tanaka.
The officials originally ruled it out but the Video Assistant Referee decided not all of the ball had gone out of play despite some replays suggesting it had.
If Japan had drawn 1-1 and not won, Germany would have gone through on goal difference.
It was the first time in their illustrious World Cup history that Germany had failed to win any of their first two group matches and despite bagging a win on their last attempt, they will depart Qatar.
"For me personally this is an absolute catastrophe," Germany forward Thomas Muller, 33, told ARD, suggesting he will now retire from international football.
"If this was my last game, I want to say a few words to our fans who have supported me for years. I have always tried to leave my heart on the field."
They were teetering on the brink of elimination going into the game after losing their opening group match to Japan and then snatching a 1-1 draw against Spain.
Storied Germany are the biggest casualties of the competition along with second-ranked Belgium, who went out earlier on Friday.
Coach Hansi Flick took over from 2014 World Cup-winning coach Joachim Low last year after a disappointing last-16 exit at the Euros.
Flick made a great start, winning his first eight games in charge but results in the Nations League then dropped off for the Germans.
Costa Rica, who required at least a point to have a chance to progress, rarely crossed into the German half, with only about 25 per cent possession in the first 45 minutes.
