Isabella County Sheriff Leo Mioduszewski said the man was identified as Eric Ramsey, 30, of Mount Pleasant.
"We don't know what possessed him to do that. We may never find out," Mioduszewski said.
The woman was abducted from campus in her own vehicle Wednesday night and taken to a home off campus where she was bound and raped. The sheriff said Ramsey then put her back in the Ford Escape and pledged to kill her, but she escaped from the moving vehicle and ran to a home yelling for help.
While the woman was inside talking to an emergency dispatcher on the phone, Ramsey "ended up pouring gasoline on the house and then lit it on fire," Mioduszewski said in a statement.
Early Thursday morning, Ramsey was spotted in Otsego County, where he rammed the first of two state police cars. The sheriff said he subsequently stole a truck and was fatally shot by a deputy in Crawford County, 85 miles north of the university.
The fire at the Mount Pleasant home was extinguished by the owner.
Campus police Chief Bill Yeagley said Ramsey told the woman that he chose her at random outside the Student Activity Center on campus. The chief said the woman saved her own life by fleeing from the car.
"I believe she made all the right choices," Yeagley said. "She's the true hero in this."
Central Michigan President George Ross said the school would support the Grand Rapids-area woman and her family.
Ramsey has been on parole since last summer after serving the minimum five-year prison sentence for assault with intent to do great bodily harm, according to Corrections Department online records. The maximum sentence was 15 years. Inmates are eligible for a parole review once they serve the minimum punishment.
"The parole board generally doesn't give a rationale for why or why they don't parole (an inmate)," said Russ Marlan, a Corrections Department spokesman. "I looked at his behavior in prison. He was pretty good. He had six misconducts over five years. That's a small amount. He was in minimum security the entire time."
Ramsey had a job, regularly met with his parole officer and had tested negative for drugs, Marlan said.
His record also included convictions for destruction of police or fire property, resisting police and assault with a dangerous weapon.