The prisoners fled the prison in Ciudad Victoria, the capital of Tamaulipas state, through a tunnel 40 meters (130 feet) long and five meters (16 feet) deep, the state security service said on Twitter.
Twelve of them have since been recaptured, it said.
Fugitive inmates killed one person in the process of stealing a getaway car, state security spokesman Luis Alberto Rodriguez told a local radio station.
Tamaulipas, which borders the US state of Texas, is a hotbed of organized crime because of its strategic location on the drug trafficking routes to the United States.
The Mexican state is the scene of a bloody turf war between the Zetas and Gulf cartels.
Rodriguez said the tunnel had been dug in the back of the prison, an area that has been under the de facto control of the inmates "for many years," he told radio station Formula.
The escape comes one week after five members of the powerful Sinaloa drug cartel escaped from another prison in northwestern Mexico.
They included Juan Jose "El Negro" Esparragoza, the son of one of the cartel's founders.
The Sinaloa cartel's longtime leader, Joaquin "El Chapo" Guzman, notoriously escaped from prison twice -- once in 2001 and again in 2015 -- before being recaptured last year and extradited to the United States, where he is imprisoned in solitary confinement awaiting trial.
