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The Blues were 2-1 down from the first leg in Germany last week but turned the tie around at Kingsmeadow to progress 5-3 on aggregate.
Chelsea will play Barcelona in the final in Gothenburg on May 16 and are still capable of winning an unprecedented quadruple under Emma Hayes this season.
Fran Kirby put Chelsea ahead in the tie on away goals in the first half but Bayern restored the initiative through a spectacular strike from Sarah Zadrazil.
Ji So-yun made it 3-3 on aggregate and the tie was heading to extra-time before a late goal from Pernille Harder had Chelsea going through.
As Bayern pushed desperately for an equaliser in stoppage time, Kirby ran half the length of the pitch to wrap up victory in a dramatic finish to an astonishing game and spark wild Chelsea celebrations.
Chelsea are the first English side since Arsenal in 2007 to reach a European final.
Before facing Barcelona, Chelsea could wrap up the WSL title with wins over Tottenham on Wednesday and Reading on Sunday.