Sherelle Roberts of Lexington police said Cleary's vehicle hit a parked car and was finally pulled over in front of police headquarters.
The driver was intoxicated, she said.
Cleary, one of eight children, is a director of the New Zealand company Clear Mountain.
He faces several charges, including criminal mischief in the first degree, fleeing and evading police, and drink-driving.
He was booked at the Fayette County Detention Centre after paying US$10,000 ($12,300). He listed his address as Clearsky Farms on Russell Cave Rd. A worker there said they had no comment on the arrest.
His father, Irish-born Eamon Cleary, became one of New Zealand's richest people, with an estimated wealth of $1.2 billion in the NBR Rich List in 2012. He made his fortune buying land and converting it to dairy farms in the 1990s. He died in September last year.