A soldier read the statement aloud, which was signed by Lieutenant Colonel Paul-Henri Sandaogo Damiba, on live television. Earlier reports had said that soldiers had surrounded the building of the state-run broadcaster RTB and were planning to air a message.
A captain, who said he represented the Patriotic Movement for Safeguarding and Restoration, said this group would work on a timetable for holding new elections that was "acceptable to everyone."
A military junta also runs neighboring Mali after a pair of coups in late 2020.
The West African country was plunged into turmoil after a group of soldiers launched a mutiny at a barracks in the capital Ouagadougou on Sunday. They then detained Kabore at his house on Monday.
Earlier in the day, the president called on "those who have taken up arms to abandon them" in a tweet.
"What appeared to be a simple mutiny launched by some elements in the army on 23 January is evolving, hour by hour, into a military coup against our hard-fought democracy," his party, the People's Movement for Progress (MPP) said in a statement.
Kabore's party also said the president had survived an "aborted assassination attempt."
The military did not say where they were holding Kabore, but said the seizure of power had been carried out "without any physical violence against those arrested, who are being held in a safe place, with respect for their dignity.'
UN Secretary-General Antonio Guterres slammed the military's move. He "strongly condemns any attempted takeover of government by the force of arms," a spokesperson said.
Guterres also called on the military "to ensure the protection of the physical integrity of the president and of the institutions of Burkina Faso."
French President Emmanuel Macron said his country stood with the Economic Community of West African States (ECOWAS) "in condemning this military coup."
Macron added that this "is the latest in a succession of several military coups that are very worrying at a time when the region's priority should be the fight against Islamist terrorism."
US State Department spokesman Ned Price called for the president's release. "We call for the immediate release of President Kabore and other government officials and for members of the security forces to respect Burkina Faso's constitution and civilian leadership," Price told a media briefing.
Why is there unrest in Burkina Faso?
Like its neighbor Mali, Burkina Faso has also been suffering from attacks related to various insurgent Islamist groups that are active across the Sahel region.
Burkina Faso had initially been spared the conflict seen in Mali, Niger, Mauritania and Chad, but in 2016 gunmen killed at least 30 people in an attack on a hotel and restaurant in the capital.
Since then the attacks have continued. In June 2021, an attack on the northeastern village of Solhan killed over 130 civilians.
The West African country is also one of the poorest in the world and the conflict has sparked a humanitarian crisis and popular unrest.
The last time the country was subjected to a successful coup was in 1987 when Blaise Compaore took power. He held onto his position through four election victories. However, he was forced from power in 2014 by street protests, fleeing to the Ivory Coast.
Kabore, who had previously served as prime minister, was elected president in 2015.