The Blues - nine time winners of the competition - went ahead just after half-time through Charlie Webster.
But it was Forest who would score next and eventually win the match in the most dramatic fashion.
With four minutes to go, Forest equalised against the run of play through Dale Taylor.
Detlef Esapa-Osong then put Forest ahead after a scramble in the box before scoring his second to put the game to bed.
Three goals in seven minutes for Warren Joyce's Forest, and they will now meet Manchester United in the final.
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