Alejandro Garza, a Nuevo Leon state prosecutor, said Cavazos was "leading the front and showing his face in the fight against organised crime".
Garza said the mayor was taken from his home around midnight by men wearing uniforms from a police agency that was dissolved years ago.
Spate of violence
At least six people were shot dead after gunmen opened fire indiscriminately at a house party in a low-income neighbourhood in Cuidad Juarez on Sunday, witnesses have said.
In a separate attack gunmen killed three women and a man at another party.
On the same day, the bound, bullet-ridden bodies of four men were discovered in a ravine on the outskirts of the border city.
In the state of Oaxaca police said eight men including a teenage boy were out on a hunting trip in a rural area when they were attacked, and their bodies were found on Sunday piled up in a pickup truck.
Meanwhile in violence at the weekend, attackers threw grenades at offices of the Televisa network in Monterrey and the border city of Matamoros. No injuries were reported.
In the Gulf coast state of Veracruz, police said on Sunday they found the bound, burned remains of a body with a federal police badge.
Drug violence has claimed more than 1,400 lives in Ciudad Juarez – and 28,000 nationwide – since the government stepped up its offensive against drug cartels in late 2006.