Crews managed to rescue 108 people from the Virgo Transport 8 ship and were still searching for any other survivors, the agency added.
At a rescue centre, passenger Rinto Arahab said he held onto the ship's railings after it flipped over, then managed to climb to a higher point on the stricken vessel as it tossed and turned in the water.
"I felt resigned," he told Reuters in the South Kalimantan city of Banjarmasin, where the ship had been heading before it lost contact with authorities.
"I could only pray for my safety."
Families of passengers also gathered in the city, waiting for news.
One woman, who gave her name as Putri, said she had been on the phone to her husband Herman who was on the boat before they lost reception.
"Then after that there was no more contact, we couldn't reach him," she said.
The ship set sail with 243 people, including 30 crew members, from the East Java city of Surabaya on Saturday, the search and rescue agency said.
A passing helicopter reported the ship had capsized, transport minister Dudy Purwagandhi told a press conference.
The cause was still being investigated, he added.
The search-and-rescue agency earlier said the ship had sunk.
Rescued passengers had been taken to other ships and some of them would be taken to an East Java port, I Putu Sudayana, the head of the agency's Banjarmasin office, said in a video statement.
The 119-metre Virgo Transport 8 ferry was built in Japan in 1987 and operated there until last year before being sold to the current Indonesian operator, PT Virgo Karya Shipping.
Indonesia, an archipelago of 17,000 islands, relies heavily on ferries for transportation, with sea routes more affordable and readily available than air travel.
with AP
15°C