The destruction of the facility on Tuesday unleashed mass flooding, forcing thousands of residents to flee and wreaking environmental havoc.
Russia and its proxy officials in Ukraine have blamed Kyiv for destroying the dam but have offered varying explanations.
The Security Service of Ukraine (SBU) posted a one-and-a-half minute audio clip on its Telegram channel of the alleged conversation, which featured two men who appeared to be discussing the fallout from the disaster in Russian.
"They (the Ukrainians) didn't strike it. That was our sabotage group," said one of the men, who the SBU described as a Russian soldier. "They wanted to, like, scare (people) with that dam."
"It didn't go according to plan, and (they did) more than what they planned for."
The SBU did not offer further details of the conversation or its participants. It said it had opened a criminal investigation into war crimes and "ecocide".
"The invaders wanted to blackmail Ukraine by blowing up the dam and staged a man-made disaster in the south of our country," the SBU said in a statement.
Meanwhile, air strikes were being traded from both sides on Friday.
At least one person was killed in a combined assault of cruise missiles and attack drones, Ukrainian authorities said.
The Ukrainian military reported shooting down four out of six missiles launched during the attack, which the air force said lasted around six hours, and 10 out of 16 drones.
The interior ministry said one person had been killed, three were wounded, and four buildings were destroyed from falling debris.
It posted images on the Telegram messaging app of firefighters attending to the smouldering wreckage of what appeared to be residential homes.
The air force also said two cruise missiles had struck a civilian object in the central Ukrainian region of Cherkasy during an earlier attack on Thursday evening.
Regional governor Ihor Taburets said at least eight people had been wounded in that strike, which he said hit a carwash and an industrial object.
And Russia says three people were hurt by shards of broken glass on Friday when a drone struck a residential building in the southern Russian city of Voronezh.
Photos posted on social media showed damage and scorch marks to the facade of the apartment block.Â
Regional governor Alexander Gusev said the three people had received medical attention on the spot and declined hospital treatment, suggesting their injuries were light.
Drone strikes inside Russian territory have become a frequent occurrence, but they more typically happen in areas such as Belgorod that lie closer to Ukraine. Voronezh is about 180 km from the Ukrainian border.
Ukraine does not comment on alleged cross-border attacks into Russia.
Russia accused Ukraine last month of launching two drones at the Kremlin in what it said was an attempt to kill President Vladimir Putin. Kyiv denied involvement in that incident.
