The message, which is in Spanish, said that the collective is "tired of the criminal group the Zetas, which is dedicated to kidnapping, stealing and extortion", reports BBC News.
It also warned that the cartel had "made a great mistake" in abducting one of its members in Veracruz, a port city on the Gulf of Mexico.
The video features a person dressed in the Guy Fawkes mask, also seen in Alan Moore's comic book series V for Vendetta, that has become associated with global activist group.
The person says that the group knows the identities of people, including police officers, journalists and taxi drivers, who assist the cartel in its business.
Anonymous warned that it will publish photographs and others details unless the kidnapped member of its order, who is yet to be identified, is released.
"We cannot defend ourselves with weapons, but we can with their cars, houses and bars," the message added.
"It's not difficult. We know who they are and where they are."
The video, posted on October 6 under the username MrAnonymousguyfawkes, was picked up at the end of last week by the global intelligence thinktank Stratfor.
Statfor said that if Anonymous carried out its threat, the people identified would most likely be murdered by rival gangs, while there could be reprisal attacks against suspected hackers.
In September, the bodies of a man and woman were found hanging from a bridge in Nuevo Laredo in Mexico. A message attached to them read: "This is going to happen to all of those posting funny things on the internet."
Anonymous sprung to fame this year after claiming responsibility for a number of attacks, including hacking the websites of The Sun and a site run by technology giant Apple.
Today, the boss of a UK intelligence agency warned that cyber attacks on the government, companies and individuals are at "disturbing" levels.