Activists alleged the death toll was higher but did not provide a figure.
A provisional assessment showed 24 inmates were fatally shot by "warning" shots from guards as they tried to escape from the Makala Central Prison in the capital of Kinshasa early on Monday, Congolese Interior Minister Jacquemin Shabani said on the social platform X.
"There are also 59 injured people taken into care by the government, as well as some cases of women raped," he said, adding that order had been restored at the prison, part of which was burned in the attempted jailbreak.
He did not elaborate on the incidents of rape. The prison holds both male and female inmates.
It was not known if all 129 fatalities were inmates and it was not immediately clear how the stampede occurred.
Makala, Congo's largest penitentiary with a capacity for 1500 people, holds more than 12,000 inmates, most of whom are awaiting trial, Amnesty International says in its latest country report.
The facility has recorded previous jailbreaks, including in 2017 when members of a religious sect stormed the prison and freed dozens of inmates.
Gunfire inside the prison started about midnight on Sunday and lasted into Monday morning, residents in the area said.
Videos posted online show bodies lying on the ground inside the prison, many of them with visible injuries.
Another video shows inmates carrying bodies that appeared to be lifeless onto a vehicle.
There were no signs of forced entry into the prison, which is in the city centre, 5km from the presidential palace.
The attempted escape was plotted from inside the prison by inmates in one of the wings, Mbemba Kabuya, the deputy justice minister, said.
In the hours after the attack, officials visited the prison as authorities convened a panel to investigate the incident.
Calling for an independent investigation, rights groups and the opposition accused the government of using excessive force and covering up the true death toll.
Martin Fayulu, an opposition leader, compared the death toll to "summary executions" and said it was an "unacceptable crime that cannot go unpunished".
Makala - among other prisons in Congo - is so overcrowded that inmates often starve to death, activists say.
Scores of prisoners have been released in recent months as part of efforts to reduce the number of inmates.
Justice Minister Constant Mutamba called the attempted jailbreak a "premeditated act of sabotage" and promised a "stern response".
His deputy, Samuel Mbemba Kabuya, blamed the country's magistrates and judges for the overcrowding in prisons, saying people were quickly jailed at the early stage of their trials.
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