After driving into a group of people, Abdul Razak Ali Artan then jumped out of the vehicle, stabbing people with a butcher's knife at the school's Columbus campus.
Eleven people were injured in the attack before the suspect was shot dead by a police officer.
Two US government sources say the assailant was an 18 year-old immigrant from Somalia and a lawful permanent resident of the United States.
Other sources say Mr Artan might have been older, possibly 20.
US Congressman Adam Schiff, a California Democrat and the ranking member of the House Permanent Select Committee on Intelligence, has issued statement saying the incident bears all of the hallmarks of a terrorist attack carried out by someone who may have been self-radicalised.
Of the people hurt in the attack, one was critically injured while 10 others also sustained injuries.
Despite the seriousness of the attack, Ohio's Governor John Kasich says the situation could have been much worse.
"Think about what this incident, what this tragedy could have meant. A man with a butcher knife and who knows what else, I don't know what else he may have had. Think about driving a car into a bunch of people. Think if you were standing there what the terrible stuff you would see. Fortunately, from what I'm understanding from both doctors here, that they expect a full recovery.
Before the gunman was shot dead, the university had issued a series of tweets warning students that there was an active shooter on campus near the engineering building and that they should run, hide or fight.
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.
Angshuman Kapil, a graduate student, was outside the building when the car drove onto the sidewalk.
"I heard a car, you know when a car brakes with a screeching sound. So suddenly a car appeared on the sidewalk and it just hit everybody who was in front. So I saw one guy flipped over because of the impact. After that everybody was shouting, 'Run! Run! Run!'"
ABC News in the US is reporting that minutes before the attack started, Mr Artan left a post on what appeared to be his Facebook account which appears to have since been disabled.
Media reports say the post read "America! Stop interfering with other countries, especially Muslim Ummah (community). We are not weak. We are not weak, remember that. If you want us Muslims to stop carrying out lone wolf attacks, then make peace. We will not let you sleep unless you give peace to the Muslims."
Mr Artan was a third-year logistics management student who had recently transferred to Ohio State from Columbus State University.
Leaders of mosques and Muslim organisations in Ohio have strongly condemned the attack, including Ohio's local mosque president Basil Gohar.
"As Ohioans, as Americans and as Muslims, we are horrified by this tragedy and this attack. We condemn it in the strongest terms, and we stand here as Muslims to say: 'This has nothing to do with our faith,' and we stand with our Somali brethren that it has nothing to do with their culture or their heritage."
US government sources say investigators believe the attacker was a Somali refugee.
