The suspect, identified as Kendrex White, was apprehended about two minutes after campus police received reports of the attack on the school's main grounds. White was being questioned by police and formal charges related to the attack were likely to come later, Reuters reported.
"I don't know what his motivation is," University of Texas at Austin Police Chief David Carter told a news conference.
White has been booked by Austin police on a single charge that was not listed in online jail records.
All the victims were found in about a one-block area and were men aged 20 or 21, police said. Their names have not been released.
"There are no words to describe my sense of loss," University President Greg Fenves told the news conference.
The person killed was found dead at the scene, Austin-Travis County EMS Captain Rick Rutledge said in a telephone interview.
The university canceled classes for the day.
“Our prayers go out to all those affected by today’s tragic events,” Texas Governor Greg Abbott said in a statement.