"We got him," Cox told reporters.
The suspect, identified as Tyler Robinson, had confessed to a family friend -or "implied that he had committed the murder" to that friend - and that person in turn had contacted the Washington County sheriff's office on Thursday.
A family member interviewed by investigators said Robinson had become more political recently and spoke in a disparaging manner about Kirk, Cox told reporters.
Robinson was taken into custody on Thursday night, about 33 hours after Kirk's murder, FBI Director Kash Patel said at the press conference.
Kirk, a close ally of US President Donald Trump, was killed by a single bullet as he spoke onstage at an outdoor amphitheatre at Utah Valley University on Wednesday.
Previously, US investigators said they had found the bolt-action rifle believed to have been used to kill Kirk and released images of a person of interest.
Investigators spoke to Robinson's roommate, who showed them comments Robinson had made on Discord, a chat and streaming platform popular with gamers, discussing retrieving a rifle from a drop point and then leaving the rifle in a bush wrapped in a towel.
This matched the description of the gun that authorities recovered after the shooting in a wooded area near campus.
Ammunition found at the scene had been inscribed, Cox said.
The messages on the casing included: "What's this;" "Oh, Bella Ciao, Bella Ciao, Bella Ciao, Ciao;" "If you read this, you are gay, LMAO;" and "Hey fascist, catch!," Cox told reporters.
Cox, a Republican, called Kirk's killing an "attack on the American experiment" and he urged a new generation to "choose a different path".
Robinson is believed to have acted alone, and the investigation is ongoing, Cox said.
Kirk, a well-connected activist, author and podcast host, helped build support for Trump and the Republican Party among younger voters.
Kirk was the co-founder and president of the conservative student group Turning Point USA and appeared at Utah Valley on Wednesday as part of a planned 15-event "American Comeback Tour" of US college campuses.
His killing stirred outrage and denunciations of political violence from Republicans, Democrats and foreign governments.
Kirk frequently engaged with his critics from the far left to the far right, often inviting members of his audiences to debate him live.
with AP
