"The suspected murderer of the three sisters is in our hands," Colonel Helmegilde Harimenshi said, adding the suspect had "confessed".
The suspect reportedly acted alone, Harimenshi said.
Initial reports said the man who carried out the first attack killing two nuns on Sunday afternoon fled the scene.
The second attack in the convent took place in the early hours of Monday morning.
The man arrested, who has not been named, was found with a key to the convent and a mobile telephone belonging to one of the nuns, according to police sources.
The three Roman Catholic nuns were named as Olga Raschietti, 83, Lucia Pulici, 75, and Bernadetta Boggian, aged 79.
Police said two nuns were stabbed to death, with the killer then battering one of the two with a rock.
Hours later, before dawn on Monday, another nun in the same convent was killed, her body beaten and head hacked off.
Pope Francis on Monday issued his condolences.
"The Holy Father begs the Lord to welcome into his kingdom of peace and light these three faithful and devout nuns," he said in a statement.
Burundian Vice-President Prosper Bazombanza has said the government was "appalled by such barbarity".
In 2011, a Croatian nun and an Italian charity worker were killed in an apparent botched robbery in northern Burundi.
The small nation in Africa's Great Lakes region emerged in 2006 from 13 years of brutal civil war and its political climate remains fractious ahead of presidential polls due in June 2015.