The Chelsea forward's double was in vain as his team's 4-2 win was not enough to secure a knockout spot in Group E, with Japan's 2-1 triumph over Spain sending both nations through instead.
It marks the end of another dour World Cup adventure for Germany, who failed to keep a clean sheet for the 12th major tournament match in a row.
Havertz, who came off the bench to spark hope, acknowledged the pain and frustration of seeing his side come up short, despite playing all their cards in their final game.
"It is bitter for all of us," he stated. "We said at half-time that we shouldn't take it easy, that anything can happen. We didn't expect that to happen. It's like a horror movie.
"We saw the live table in the stadium, we recognised the situation from spectators and the reactions on the bench.
"There was still hope, because we assumed Spain would score another goal. Then we saw on the screen that their match was over, and that was it."
Germany's exit in Qatar marks the second successive World Cup where they have failed to get out of the group stages, and coupled with a last-16 loss at Euro 2020, makes for three tournaments on the bounce with poor results.
With the nation set to host European Championship in just over 18 months' time, Havertz understands the task of reigniting pride in their team will require a degree of soul-searching.
"If three tournaments in a row, you go out early, it makes you wonder what is going wrong," he added. "We need to change something. We are aware of that.
"When it counts, we didn't perform. We need to be honest with ourselves. Over the past five years, things haven't gone right, and it is our own fault. We are to blame."
Head coach Hansi Flick, in his first major tournament since succeeding Joachim Low, praised Havertz's contribution, but stressed his disappointment at their missed chances in Qatar.
"If I look at positives, Kai turned the game around," he added. "We can leave the pitch and go home with a victory. But if you look at the matches, it is our fault.
"We had chances against Spain and against Japan. You have to take them and score goals. There were lots of individual mistakes, and that is what makes me angry."