Police said they are investigating whether it was a terrorist attack. Nine people were hurt, one critically.
The attacker was identified as Abdul Razak Ali Artan, a Somali-born legal permanent resident of the US, according to a US official who wasn't authorised to discuss details of the case publicly and spoke on condition of anonymity.
The FBI and other agencies joined the investigation.
The details emerged after a morning of confusion and conflicting reports that began with the university issuing a series of tweets warning that there was an "active shooter" on campus near the engineering building and that students should "run, hide, fight."
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Numerous 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.
Ohio State Police Chief Craig Stone said that the assailant deliberately drove over a curb outside a classroom building and then got out and began attacking people with the knife. A campus officer who happened to be nearby because of a gas leak arrived on the scene and shot the driver in less than a minute, Stone said.
"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 that the man didn't say anything when he got out.
Asked at a news conference whether authorities were considering the possibility it was a terrorist act, Columbus Police Chief Kim Jacobs said: "I think we have to consider that it is."
The Islamic State group has urged sympathisers online to carry out attacks in their home countries with whatever weapons are available to them.