Bayern, four-times winners, were stunned and extra time could not separate the sides despite Bayern earning a penalty which Arjen Robben had saved by Cech.
Chelsea lost to Manchester United in the 2008 final on penalties but erased that painful memory as they held their nerve to become London's first European champions.
"I believe a lot in destiny. It was written a long time ago. This team is amazing and I dedicate this cup to all the managers and players we had before," Drogba said.
"(My equaliser) changed the game. Life is fantastic."
Bayern were left to rue a third near miss this season after finishing runners-up in the Bundesliga and the German Cup but this was the harshest of blows.
Image: Chelsea's Juan Mata holds up the trophy at the end of the Champions League final soccer match between Bayern Munich and Chelsea in Munich, Germany Saturday May 19, 2012.