The Animal Plant and Health Inspection Service is the U.S. Government agency that regulates the entry of non-working dogs (pets, show dogs, etc.) and cats into the United States. The Public Health Service currently lists Chile as a country that has rabies.
Please see the following link on the APHIS website for further Information on entering live animals into the U.S.
APHIS’s counterpart in Chile is Servicio Agricola Ganadero (SAG). They have very stict entry and exit requirements for pets, so make sure you obtain all the required paperwork before your trip.
Please see the following link on the SAG website for further information.
It may also be a good idea to contact your airline as well to see if they have any special transportation requirements.