Some of the passengers were exchange students from South Korea. Others were from Canada and Washington state, hospital officials said.
One survivor, Yoo Byung Woo, 25, told the Oregonian that he and other passengers thought the bus driver wasn't driving as slowly as he should have been for the conditions.
"I felt like he was going too fast," Yoo said. "I worried about the bus."
Yoo said it was snowing and foggy as the bus traveled west. One of the riders, who was frightened, asked if they could take another route, Yoo told the newspaper. Some passengers were dozing when the driver slammed on the brakes.
Rocks smashed through windows after the bus crashed through a guardrail and rolled down a slope, Yoo said.
The NTSB said the 1998-model bus rolled at least once.
A hospital official said it appeared 46 people were aboard the bus, and the 37 survivors were sent to hospitals.
Fourteen of those aboard were at St. Anthony Hospital in Pendleton on Monday morning, one in serious condition, said spokesman Larry Blanc. Seven were discharged Sunday.
Blanc said 16 people were sent to other hospitals in the region, including the Oregon Health & Science University hospital in Portland.
Umatilla County Emergency Manager Jack Remillard said the bus was owned by Mi Joo Tour & Travel in Vancouver, B.C. A bus safety website run by the U.S. Department of Transportation said Mi Joo has six buses, none of which have been involved in any accidents in at least the last two years.
The bus driver was among the survivors but had not yet spoken to police because of the severity of his injuries.
Another passenger, who declined to give his name, told the Associated Press he was asleep when the bus went out of control.
"Suddenly people were screaming and the bus [went] down the hill," said the 22-year-old passenger. "I woke up. I feel I'm dying. I grab the seat. Finally the bus stopped."
The man, who lives in Seoul, said he has been studying English at a Vancouver university since November. He and five friends joined a nine-day bus tour of the western United States that included stops in Seattle, Portland, San Francisco, and Las Vegas.
All of his friends survived.
More than a dozen rescue workers descended the hill and used ropes to help retrieve people from the wreckage in freezing weather.
The NTSB said two investigators were expected to arrive at the crash site Monday. It said they will be looking into why the bus left the road, the condition of the road at the time, the condition of the guardrail, and operations of the company that owns the bus.
A spokesman for the agency, Peter Knudson, said seatbelts aren't required on such buses. "We have been concerned about this for some time," Knudson said.