As the flame was kindled in the 2,600-year-old Temple of Hera at Ancient Olympia, International Olympic Committee president Thomas Bach and Rio organisers sought to rally support.
Bach said the Rio Games will be held "in a world shaken by crises" but hailed preparations for the first Olympics in South America.
"Brazilian people will enthusiastically welcome the world and amaze us with their joy of life and their passion for sport," he predicted.
Brazil's President Dilma Rousseff cancelled her attendance as she battles threatened impeachment proceedings.
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In her place, Rio 2016 organising committee chairman, Carlos Nuzman, said the flame "brings a message that can and will unite, our dear Brazil."
"Brazil is ready," Brazil's acting sports minister Ricardo Leyser told reporters after the ceremony.
Setting off from a country at the forefront of Europe's migrant crisis, the flame will tour Greece and then go to scores of Brazilian cities arriving at Rio's Maracana Stadium for the August 5 opening ceremony.
Petrounias passed it to Brazilian volleyball legend Giovane Gavio, an Olympic champion at the 1992 Barcelona and 2004 Athens Games.
Overall, some 12,000 torchbearers will carry the flame through Brazil.
In a gesture to the migration crisis, a Syrian man who lost a leg in his country's civil war will carry the torch through the Eleonas refugee camp in Athens next Tuesday.
The man, whose is to be named Friday, has been living and working in Athens since being granted asylum in Greece.
Brazil's preparations have been overshadowed by the government crisis caused by accusations that Rousseff juggled government accounts to disguise budget shortfalls during her 2014 reelection.
The beleaguered president insisted this week that Rio's preparations are ahead of schedule.
"The Games are in a totally adequate situation, we're even a little ahead of schedule and more so than we had planned for," Rousseff said Tuesday.
"(In terms) of the organisation of the Games everything was done...I don't think it's really a problem for the Games," said Leyser.
Bach on Wednesday said he was confident that Rio would lay on an "excellent Games".
"Ninety-eight percent of all infrastructure is ready, the test events are going successfully...We are just full of anticipation to come to Brazil," he told reporters on Thursday.
The IOC and Rio organisers have played down crime concerns and fears over the quality of water in Rio bay to be used for yachting and some swimming.
"We are very confident that the competition area will offer safe and fair conditions," Bach said this week.
Greek Olympic Committee chairman Spyros Kapralos said Brazil would pull through, much like Athens overcame chronic delays before it hosted the 2004 Games.
"I think like we had in Athens, the same will happen in Rio, everything will come together and we will experience a fantastic Games," he said.