The operation took place on Sunday in Montanita in the southern department of Caqueta.
There were "11 dead and two wounded, including a minor who had been recruited by force," the minister told local media.
The 13 were part of a faction commanded by a former FARC rebel leader, Rodrigo Cadete, who rejected the 2016 peace agreement with the government of President Juan Manuel Santos.
The minister said the faction had been making threats against the mayor of the Caqueta capital of Florencia as well an energy company in the region.
"The criminals had been demanding extortion payments from businesses" in Florencia and its surrounding area, the army said in a statement.
Under the peace accord, the Revolutionary Armed Forces of Colombia (FARC) disarmed its 7,000 fighters in order to join the political process. It is now a political party.
However, remnants of the rebel force said by the army to number about 1,200 are still active in drug-trafficking in the southern border areas.