The Tartan Army went into Sunday's pivotal showdown in Stuttgart aware that they realistically required all three points to progress in second in Group A or as one of the four best fourth-placed sides.
Despite all their possession during the first half, Scotland failed to have a single shot. It was the second time in three matches at this tournament that they have recorded the unwanted feat.
After the break, Scotland became increasingly desperate and no more likely to score, whereas Hungary squandered several opportunities to put the contest beyond their opponents.
Kevin Csoboth netted the all-important goal in the final minute of 10 added-on after Scotland committed to trying to find the goal that never looked like arriving on a deflating night for their travelling support.
Hungary must now wait to see whether they progress to the last 16, but Scotland and Clarke are left to reflect on a disappointing stay in Germany.
What did Clarke have to say?
The game's biggest talking point came when Stuart Armstrong went down under a challenge from Willi Orban during the closing stages, the referee immediately waving away Scotland's appeals.
Replays suggested that there had been contact from the Hungary defender, although Armstrong did hold onto the shirt of his marker before going down.
Clarke told BBC Sport: "The goal comes as a consequence of us trying to win the game.
"The goal is almost irrelevant. The moment for me was the penalty. It was 100% a penalty. Someone has to explain to me why it wasn't a penalty because otherwise I'm thinking there's something wrong.
"I don't understand how VAR can look at that and say it's not a penalty."
Varga update
Earlier in the second half, there were extremely worrying scenes as Hungary forward Barnabas Varga was involved in a sickening collision when challenging for a high ball.
There was immediate concern from his teammates who shouted for medical staff to attend the situation in the penalty area, with sheets being held around Varga as he received treatment.
A statement from the Hungarian Football Federation has provided an update on the situation, saying: "Barnabas Varga's condition is stable.
"The Ferencvaros player is currently in one of the hospitals in Stuttgart. We will inform you immediately if there is any news about his status."
Varga has started each of Hungary's three group games, scoring in the opening fixture against Switzerland.