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6 year oldCristhian Bahena Rivera, 24, is in the US illegally, authorities say.
Remains believed to be Ms Tibbetts's were earlier found in a cornfield near her hometown of Brooklyn, Iowa.
The 20-year-old University of Iowa student went missing on 18 July while dog-sitting at the home of her boyfriend and his brother.
She was last seen by neighbours on a run. Her family reported her missing the next day when she did not arrive at work.
Iowa Division of Criminal Investigations special agent Rick Rahn said in a press conference Mr Rivera is an undocumented immigrant.
Mr Rivera allegedly saw Ms Tibbetts running and followed her before he says he "blacked out".
Mr Rahn said Mr Rivera was "very compliant" and willing to talk, and said that he led police to her body.
An arrest warrant says Mr Rivera is charged with murder in the first degree. He would receive a mandatory life sentence if convicted.
Authorities were led to the body in the early hours of 21 August, in a cornfield in Poweshiek County - which includes Ms Tibbetts's hometown, Brooklyn.
The body had corn stalks placed on top of it, Mr Rahn told reporters.
A surveillance video of Ms Tibbetts jogging, and of Mr Rivera's car, helped authorities track the suspect down, he said.
Police received more than 4,000 tips during their 34-day investigation.
Investigators had previously used social media and Fitbit data in an effort to trace her.
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