Vice-President Prosper Bazombanza told reporters the blaze hit an overcrowded facility in the capital Gitega, with at least 69 people seriously injured.
Images circulating online show a building engulfed in flames and piles of bodies, said to be of inmates in the jail.
“It is really catastrophic,” one inmate told the BBC over the phone.
“I’d say almost 90% of sleeping halls are burnt.”
A journalist outside the prison told the BBC that nurses from Gitega hospital have entered the prison to help the victims, saying they are taking out the dead and wounded.
The blaze started at around 04:00 local time (02:00 GMT).
“We started shouting that we were going to be burned alive when we saw the flames rising very high, but the police refused to open the doors of our quarters, saying ‘these are the orders we have received’,” one inmate told AFP news agency over the phone.
“I don’t know how I escaped, but there are prisoners who were burned completely,” he added.
Witnesses told AFP the fire has now been brought under control.
With capacity for 400 prisoners, Gitega prison held 1,539 inmates as of last month, according to the Christian Association Against Torture (ACAT-Burundi).
A tweet posted by Burundi’s interior ministry said a short circuit was to blame for the fire.In August, another fire broke out at the same prison, which authorities also blamed on electrical problems. That blaze caused no casualties.