The shootout took place in Villa Union, a town in Coahuila state about an hour’s drive southwest of the US border.
It comes days after Donald Trump vowed to wage war against the cartels, designating them terrorists, after nine members of the same Mormon family were killed in an attack on their convoy.
But, he added, several municipal workers remain missing and a further six police officers were injured.
The Cartel of the Northeast is said to have stormed the town of 3,000 people in a convoy of trucks, attacking local government offices and provoking a response from state and federal forces.
Ten alleged cartel members were killed in the shootout.
Some 14 vehicles were involved, Mr Riquelme added, and more than a dozen guns were seized.
Videos of the aftermath showed vehicles burned out and the facade of Villa Union’s municipal office riddled with bullets.
In other videos of the shootout, frantic people tell loved ones to stay indoors to the sound of rapid gunfire.
One features images of a damaged black pickup truck with C.D.N. written on its door – said to stand for Cartel del Noreste, or Cartel of the Northeast.
Mr Riquelme said security forces will remain in the town for several days to restore a sense of calm.
Mexico has been plagued by murders related to the drugs trade.
Officials said recently that there have been 29,414 killings so far in 2019, compared to 28,869 in the same period of 2018.
Following Mr Trump’s comments, President Andres Manuel Lopez Obrador said he would not accept any foreign intervention in Mexico to deal with violent criminal gangs.