The moon began to move between the Earth and sun at 6:19 am (2319 GMT Tuesday), and about an hour later direct sunlight was entirely blocked out in western parts of the country, authorities said.
The total eclipse was set to move in a broad arc right across the archipelago, to the eastern Maluku Islands, before sweeping out across the Pacific Ocean. Partial eclipses were also visible over other parts of Asia and Australia.
"It was very beautiful, there are a lot of people here and when the totality hit, everybody cheered. I got goose bumps."
Skies appeared to be generally clear in the most popular viewing spots, although in some areas cloud partially obscured views.
From a festival featuring live bands, to fun runs and traditional dances, events were taking place across Indonesia for an estimated 10,000 foreign visitors and 100,000 domestic tourists witnessing the phenomenon.
Hotels in the best viewing spots filled up weeks ago -- in the city of Ternate, in the Maluku Islands, officials have had to find extra space for tourists on boats.
There was blanket coverage of the phenomenon on local TV channels.
I Gde Pitana, the government's head of foreign tourism, described the phenomenon as "a tourism attraction created by God".
Many of the faithful in the world's most populous Muslim-majority country flocked to say special eclipse prayers.
"Our Prophet Mohammed said the prayer signifies the greatness of Allah, who created this wonderful phenomenon," said Ma'ruf Amin, chairman of the Indonesian Ulema Council, the country's top Islamic clerical body.
Some of Indonesia's tribespeople are fearful of the phenomenon, however. Members of the Dayak tribe in one part of Borneo island were set to perform a ritual to ensure that the sun, which they view as the source of life, does not disappear entirely.
The total eclipse will sweep across 12 out of 34 provinces in Indonesia, which stretches about 3,000 miles (5,000 kilometres) from east to west, before heading across the Pacific Ocean.
It will be visible for between just one and a half to three minutes in most places.
As well as being seen in Sumatra, the eclipse will sweep across Sulawesi and Borneo and the Malukus before heading out over the ocean.
One of the most popular events for foreign tourists will be a festival close to Palu, in Sulawesi.
Among those coming to Indonesia is Thai Princess Maha Chakri Sirindhorn, who recently hit the headlines after a $40,000 toilet custom-built for her visit to Cambodia went unused.
She will be watching the spectacle in Ternate, although officials insist that no luxury commode has been built for her on this occasion.
The last total solar eclipse occurred on March 20, 2015, only visible from the Faroe Islands and Norway's Arctic Svalbard archipelago.
Total eclipses occur when the moon moves between the Earth and the Sun, and the three bodies align precisely.
As seen from Earth, the moon is just broad enough to cover the solar face, creating a breath-taking silver halo in an indigo sky.
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