The four-time champions defeated Costa Rica 4-2 but were knocked out due to Japan's surprise 2-1 win over Spain, which saw the Blue Samurai top the group. Despite also finishing on four points, Spain progressed ahead of Germany on goal difference.
Havertz, who scored two second-half goals after coming off the bench on Thursday, said the way the players found out about their World Cup exit made it even more difficult.
"When it happens this way, it feels like watching like a horror movie," said Havertz. "We learned during the match that Japan were leading, and then the rankings were displayed in the stadium. We still had a little hope that Spain would score. But then we noticed that the Japan match was over."
Havertz blamed the team's performance against Japan for their elimination and hinted that all was not well in German football. "When you go out twice in the first round and once in the round of 16, it's extremely bitter. We have to be honest and say that for four years, everything hasn't been going well. Now we've been knocked out twice in the group stage. I don't think we're a tournament team anymore."