Eleven people were hurt, one critically.
The attacker was identified as Abdul Razak Ali Artan. He was born in Somalia and was a legal permanent U.S. resident, according to a U.S. official who wasn't authorized to discuss the case and spoke on the condition of anonymity. The FBI joined the investigation.
The details emerged after a morning of conflicting reports and confusion, created in part by a series of tweets from the university warning there was an "active shooter" on campus and students should "Run Hide Fight." The warning was prompted by what turned out to be police gunfire.
Police vehicles and ambulances converged on the 60,000-student campus, and authorities blocked off roads. Students barricaded themselves inside offices and classrooms, piling chairs and desks in front of doors, before getting the all-clear an hour and a half later.
Ohio State University police Chief Craig Stone said the assailant deliberately drove his small Honda over a curb outside an engineering classroom building and then began knifing people. Officer Alan Horujko nearby because of a gas leak arrived on the scene and shot the driver in less than a minute, Stone said.
Angshuman Kapil, a graduate student, was outside Watts Hall when the car barreled onto the sidewalk.
"It just hit everybody who was in front," he said. "After that everybody was shouting, 'Run! Run! Run!'"
Student Martin Schneider said he heard the car's engine revving.
"I thought it was an accident initially until I saw the guy come out with a knife," Schneider said, adding the man didn't say anything when he got out.
Most of the injured were hurt by the car, and at least two were stabbed. One had a fractured skull.
Columbus police Chief Kim Jacobs, asked whether authorities were considering the possibility it was a terrorist act, said: "I think we have to consider that it is."
Republican Vice President-elect Mike Pence called the episode "a tragic attack" and said "our prayers are with them all."
U.S. Rep. Adam Schiff, of California, the top Democrat on the House Intelligence Committee, said the bloodshed "bears all of the hallmarks of a terror attack carried out by someone who may have been self-radicalized."
"Here in the United States, our most immediate threat still comes from lone attackers that are not only capable of unleashing great harm but are also extremely difficult, and in some cases, virtually impossible to identify or interdict," he said.
Ohio State's student newspaper, The Lantern, ran an interview in August with a student named Abdul Razak Artan, who identified himself as a Muslim and a third-year logistics management student who'd transferred from Columbus State in the fall.
He said he was looking for a place to pray openly and worried how he would be received.
"I was kind of scared with everything going on in the media. I'm a Muslim, it's not what media portrays me to be," he told the newspaper. "If people look at me, a Muslim praying, I don't know what they're going to think, what's going to happen. But I don't blame them. It's the media that put that picture in their heads."
In recent months, federal law enforcement officials have raised concerns about online extremist propaganda encouraging knife and car attacks, easier to pull off than bombings.
The Islamic State group has urged sympathizers online to carry out lone-wolf attacks in their home countries with whatever weapons are available to them.
In September, a 20-year-old Somali-American stabbed 10 people at a St. Cloud, Minnesota, shopping mall before being shot to death by an off-duty officer. Authorities said he asked some of his victims if they were Muslim. In the past few years, London and other cities also have seen knife attacks blamed on extremists.
Neighbors said Artan was always polite and attended daily prayer services at a mosque.
Leaders of Muslim organizations and mosques in the Columbus area condemned Monday's attack while cautioning people against jumping to conclusions or blaming a religion or an ethnicity.
"It is particularly heartbreaking to see this random act of violence come to this community I hold so dear," said Ohio State graduate Nicole Ghazi, who is active in Islamic organizations.
Surveillance photos showed Artan in the car by himself just before the attack, but investigators are looking into whether anyone else was involved, police said.
The bloodshed came as students were returning to classes following the Thanksgiving break and Ohio State's football victory over rival Michigan, which brought more than 100,000 fans to campus on Saturday.
"There were several moments of chaos," said Rachel LeMaster, who works in the engineering college. "We barricaded ourselves like we're supposed to since it was right outside our door and just hunkered down."
LeMaster said she and others were eventually led outside the building and she saw a body on the ground.
Classes were canceled for the rest of the day.
The officer who killed the attacker was Alan Horujko, a member of the force for just under two years. Department of Public Safety Director Monica Moll said Horujko had done a "fabulous job."
The initial tweet from Ohio State emergency officials went out around 10 a.m. and said: "Buckeye Alert: Active Shooter on campus. Run Hide Fight. Watts Hall. 19th and College." University President Michael Drake said the warning was issued after shots were heard on campus.
"Run, hide, fight" is standard protocol for active-shooter situations. It means: Run away if possible; get out of view; or try to disrupt or incapacitate the shooter if your life is in imminent danger.
Associated Press writers Alicia A. Caldwell and Eric Tucker in Washington, Collin Binkley in Boston and Mark Gillispie in Cleveland contributed to this story.