Due to poor weather and the lack of sunshine, the flame which was lit on Sunday during rehearsals had to be used during the ceremonies.
As rain fell actors in ancient Greek costume resorted to the backup flame at the ruins of the 2,600-year-old Temple of Hera instead of a scheduled ceremony using a parabolic mirror to kindle the flame.
The torch relay will take it around Greece and on to 2018 hosts South Korea.
"Then and now, the Olympics is a symbol of hope and peace. We are carrying the ancient traditions to the future. The Games are universal and stand above all differences that divide us and are here to unite humanity," Bach said.
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"Our Olympic values have stood the test of time of peace and understanding," he added.
Bach praised the work of the South Koreans in organising the next Winter Games and added that "we are waiting for excellent Games".
"All countries have been invited to attend and it is an honour for us to host the competition. We will spread the word of peace and harmony," Lee added.
The South Korea government has promised that the Pyeongchang Games would be safe after some nations suggested they may skip the event if tensions over North Korea's nuclear ambitions worsen.
The Pyeongchang Winter Games will be held from February 9-25 only 80km from the heavily guarded border with the nuclear-armed North.
The 1988 Seoul summer Olympics took place only months after a bomb planted by North Korean agents killed all 115 people on board a South Korean plane, in an apparent attempt to scare off foreign spectators and contestants.
"From the sacred land of Olympia, we give a message of hope that all disputes and quarrels around the world would be put aside," said Olympia mayor Yiannis Kotzas.
After thanks to the god Apollo, "king of the sun and the idea of light", in the shadow of the Greek South Korean and Olympic flags, high priestess Greek actress Katerina Lehou handed the flame to the first relay runner, Greece's 24-year- old Nordic skier Apostolos Angelis.
"The Olympic flame is a universal symbol of peace and friendship," said Hellenic Olympic Committee president Spyros Capralos.
The torch relay will cover 2,129 kilometres on Greek territory and will arrive at the Acropolis in Athens on October 30.
Participating will be 505 torchbearers and 36 welcome ceremonies will be held in 20 districts during the eight days.
On October 31 the flame will be handed over to the Pyeongchang 2018 Organising Committee during a nighttime ceremony at Athens' Panathenaic Stadium, where the first modern Olympic Games were held in 1896.