The man was found dead around 3 a.m. Tuesday, sheriff's spokesman Craig Chamberlin said.
"We're not putting out the method of death," or the identities of the people involved, he said.
Law officers are obtaining warrants to search the rooms of the victim and the suspect, he said.
The sheriff's office is working with the state Department of Social and Health Services, which oversees the mental hospital located in this suburb of Spokane.
Eastern has 287 beds and serves patients from 20 Eastern Washington counties. Other counties are served by Western State Hospital at Lakewood.
The victim was found dead on the floor of the hospital's forensics unit, where patients are evaluated after being committed by a judge during a criminal case, said John Wiley, a spokesman for DSHS. People are sent there if they are considered a danger to themselves or others, he said.
Patients have been moved out of the ward where the slaying took place, DSHS said.
The union representing Eastern State workers released a statement saying workers have been "diligent in raising concerns about short-staffing" at the hospital. The workers are represented by Local 782 of the Washington Federation of State Employees.
"We have raised these issues with management for more than a year," the union statement said. "It's not clear if this understaffing contributed to this morning's tragedy."