Nine penalties in a row were scored before Chelsea substitute Romelu Lukaku had his effort saved by Bayern keeper Manuel Neuer to give the Champions League winners victory over the Europa League holders.
Fernando Torres put Chelsea ahead with a sublime first-time finish in the eighth minute before Franck Ribery equalised with a 25-metre drive in the 47th.
Eden Hazard restored Chelsea's advantage with a crisp drive in the 93rd minute and Javi Martinez dramatically levelled for Bayern with the last kick of extra-time.
Chelsea had Brazil midfielder Ramires sent off in the 85th minute for a second bookable offence.
(Writing by Tony Jimenez; in London; editing by Ed Osmond)
