Cousins is accused of shooting 21-year-old Andrew Boldt of West Bend, Wis., in the university's Electrical Engineering Building around noon Tuesday.
Cox said police are still investigating the motive behind the shooting but that it appears Cousins targeted Boldt. No one else was injured.
Cox said Boldt was a senior and worked as a teaching assistant in the school. Cousins is a student in the engineering school.
Cousins is accused of opening fire inside a basement classroom, killing Boldt and prompting officials to send a text alert to students telling them to seek shelter, police and the university said.
Cousins surrendered to a police officer within minutes of the attack, Cox said.
Students described a chaotic scene as the initial report of the shooting hit around noon on the campus in West Lafayette, about 60 miles northwest of Indianapolis.
"It was scary," said Julissa Martinez, a freshman nursing student from Portage. She was in psychology class on another part of the 40,000-student campus when she received the text message saying the university was on lockdown.
She said her professor briefly kept teaching, then stopped lecturing so that students could contact people to let them know they were safe.
"He tried to get everything under control because people were freaking out," she said, adding that students were nervous because there was a lot of speculation about the severity of the situation.
The identities of the suspect and victim were not immediately confirmed. Cox said the suspect entered the Electrical Engineering Building, "took the actions that he took" and "immediately left the facility without any other interaction that we're aware of."
Cox said the suspect wasn't immediately cooperating with investigators. He said some people witnessed the shooting, but he didn't specify whether the attack happened during a class.
Shortly after 12:03 p.m. when the shooting was reported, Purdue officials issued a text alert telling those on the campus to seek shelter. Around 1:15 p.m., the university said there was no ongoing threat on campus and allowed normal operations to resume in all buildings except the engineering facility.
Purdue officials considered the campus to be secure, said Purdue Provost Tim Sands, who in June will become president of Virginia Tech, where an April 2007 campus massacre left 33 dead.
By mid-afternoon Tuesday, Sands was encouraging students to go about their normal activities around campus, except for the Electrical Engineering Building.
But the university later announced that classes were being suspended through Wednesday. A candlelight vigil was planned for Tuesday night, with special counseling services being offered to students at several sites around campus.
Kayla Brown, a 23-year-old senior from Marion, said she was still worried even after university officials lifted the campus warning.
"I was wary to go to class but I did anyway," she said. "I didn't quite feel they could have had it all clear in that period of time."
Cox, the campus police chief, said authorities responded aggressively after the shooting was reported, with about 20 campus and city police officers at the building within minutes.
Sands said the university will offer assistance to those who need it as the circumstances of the shooting are sorted out.
"We'll provide whatever services we can to assist our students, our faculty and our staff in coming back to a sense of normality," he said.