Chelsea had gone five consecutive league games without a goal - something that hadn't happened since 1912 - and were heading for a historic low point with a sixth consecutive clean sheet after Cole Palmer missed a penalty.
But Joao Pedro's penalty in extra time broke the goal drought.
However, the losing streak in the league was not broken. Chelsea are ninth, and a Champions League spot looks increasingly distant.
Many home fans left the stadium before the match was over.
Taiwo Awoniyi was the match winner for Nottingham with two goals - the first in the second minute.
Nottingham are six points behind West Ham in the relegation zone with three rounds left to play.
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