For more than two decades, Ohio had been the only state to classify a dog "vicious" based on breed and appearance, WEWS-TV, Cleveland, reported.
Gov. John Kasich signed the measure in February.
The (Cleveland) Plain Dealer said the law removes pit bulls from the definition of "vicious" dogs.
The law also defines "vicious," "dangerous" and "nuisance" dogs without regard to breed.
The law doesn't affect community bans on pit bulls such as those in northeast Ohio.
Backers of the law said it would make it easier for dog wardens and police to protect people from all dangerous dogs, regardless of breed.
Critics, some of whom are dog wardens, point to frequent pit bull attacks.