The North said the drill is a reaction to allied exercises that it said amounted to plans for a preemptive nuclear attack by the United States.
The missile unit fired two ballistic missiles and correctly carried out its "nuclear strike mission," the General Staff of the North's Korean People's Army (KPA) said in a statement carried by the news agency KCNA.
"The KPA staged a tactical nuclear strike drill simulating scorched-earth strikes at major command centers and operational airfields of the 'ROK' military gangsters on Wednesday night," KCNA reported, using initials of South Korea's official name, the Republic of Korea.
Earlier, South Korea's military said that North Korea fired two short-range ballistic missiles into the sea, hours after the US deployed B-1B bombers for allied air drills.
North Korea has upped its provocations in recent months. The country tested dozens of ballistic weapons in 2022, including its first intercontinental ballistic missiles since 2017.
In July, North Korea test-fired a Hwasong-18 solid-fuel intercontinental ballistic missile (ICBM), in a launch that was supervised by its leader Kim Jong Un.
Last week, North Korea made a second attempt to launch a spy satellite, but failed, announcing it will make a third attempt in October.
Earlier this week, North Korean leader Kim Jong Un observed part of a drill that involved all the commanding officers and staff sections of the entire army, aimed at preparing them for an all-out war with the South.