Sergeant Pete Simpson said: “In the midst of his ranting and raving, some people approached him and appeared to try to intervene with his behaviour and some of the people that he was yelling at.
“They were attacked viciously.”
One person was pronounced dead at the scene and another died in hospital, he added.
A third person was taken to hospital with non-life threatening injuries.
It wasn’t clear why the man was shouting, Sgt Simpson said.
“He was talking about a lot of different things, not just specifically anti-Muslim.”
Evelin Hernandez, a 38-year-old resident of Clackamas, Oregon, told The Oregonian newspaper she was on the train when the man began making racist remarks to the women.
A group of men tried to quiet him and he stabbed them, she said.
Officers arrested a man who ran from the train in Portland, Oregon, reports said. Police are working to identify the man and the people who were attacked.
Sgt Simpson said the women understandably left the scene before police were able to talk with them but they would like to hear from them to help fill in what happened.
“It’s horrific,” Sgt Simpson said. “There’s no other word to describe what happened today.”