Rescue workers are struggling to reach remote parts of the province, and there are fears the number of dead may rise.
Roads have been blocked by landslides and fallen trees, and floodwaters have damaged two bridges and more than 100 houses.
More than 4,000 people have been forced from their homes, and some are sheltering in government offices.
Local residents said torrential rain began on Saturday evening and continued into the night, triggering mudslides and flash floods.
“Our house was flooded with thick mud… we immediately grabbed our valuables and ran to a neighbour’s [two-storey] house to seek refuge,” mother-of-two Lili Puji Hastuti told AFP news agency.
“It’s hard to get out of the area because many roads are blocked… I’m worried, sad and scared all at once.”
The search for victims continues in the town of Sentani, one of the worst affected areas.
At least 51 people were killed in the town, national disaster agency spokesman Sutopo Purwo Nugroho told Reuters news agency.
Another seven confirmed deaths were in the nearby provincial capital Jayapura, Mr Nugroho added.
However, a five-month-old baby was rescued in the city after being trapped under the rubble for hours, according to the military.
The baby was later reunited with his family. His father survived but his mother’s whereabouts is unknown.
Flooding is common in Indonesia, especially during the October-April rainy season.
In January, floods killed at least 70 people on the island of Sulawesi, and heavy rain in West Java earlier in March displaced hundreds.
Emergency officials say that in recent months they had warned the Papuan authorities of flood risks due to deforestation of mountains surrounding Jayapura.
Papua province borders the independent state of Papua New Guinea, to the east.