Hundreds of people were displaced after leaving submerged homes.
President Joko Widodo said he ordered his Cabinet ministers and the chiefs of the military, police and disaster agency to carry out emergency response measures as quickly as possible.
A two-year-old child was among the victims, a Reuters witness said.
She warned that the cyclone could trigger waves up to 4 meters (more than 13 feet) on Sumba, Flores and Rote islands in East Nusa Tenggara province and up to 6 meters (19.6 feet) in the southern part of the province and in the Banda Sea and Indian Ocean.
Scared residents flocked to temporary shelters across the remote region or took refuge in what was left of their homes. "Many people died here because the village was hit by both landslides and flash floods", he said, providing numbers not yet confirmed by national authorities.
"They need medicine, food, blankets".
Worst hit was East Flores regency in Indonesia's southernmost East Nusa Tenggara province.
In Australia, Labor's spokesman for global development and the Pacific, Pat Conroy, called for the federal government to respond to any requests for assistance and prepare to help with food shortages.
Images and video posted on social media showed the devastation on Lembata and Adonara islands in Indonesia as well in East Timor, also known as Timor-Leste. "We are supporting access to clean water for the centres through our existing development program", the spokesperson said in a statement.
Pounding rains challenged efforts to find any survivors. "27 people are still missing and nine people are injured", he added.
Indonesia's disaster management officials are unable to send ships to affected islands.
Images from the island showed barefoot locals wading through mud and past collapsed houses to evacuate victims on makeshift stretchers.
Fatal landslides and flash floods are common across the Southeast Asian archipelago during the rainy season.
Flash floods killed at least 84 people elsewhere and 71 were still missing in the province, local leaders said at the virtual news conference. That disaster on Lembata island killed at least 11, while at least 16 others were still buried under tons of the solid lava, Jati said.
The country's disaster agency has estimated that 125 million Indonesians - almost half of the country's population - live in areas at risk of landslides.
The country's disaster agency has estimated that 125 million Indonesians - almost half of the country's population - live in areas at risk of landslides The disasters are often caused by deforestation, according to environmentalists.