South Korea had reported that North Korean soldiers had killed the man, who had probably defected and swum into North Korean waters.
It is very unusual for North Korea to apologise to its fierce rival on any subject.
The North Korean leader said: "Comrade Kim Jong Un, the State Affairs Commission chairman, feels very sorry to give big disappointment to President Moon Jae-in and South Korean citizens because an unexpected, unfortunate incident happened," Suh Hoon, an adviser to the South Korean president, reported.
The apology comes after the South Korean government said in a statement that the man disappeared from a ship checking on possible unauthorised fishing in an area south of the boundary on Monday, a day before he was found in North Korean waters.
North Korea sent staff wearing gas masks aboard a boat near the man on Tuesday afternoon, South Korea’s Defence Ministry said. Later in the day, a North Korean navy boat arrived and opened fire at him, they added.
Sailors from the boat, putting on gas masks and protective suits, poured petrol on his body and set it on fire, the ministry said, citing intelligence gathered by surveillance equipment and other assets.
According to the North Korean message read by Mr Suh, the country's troops first fired blanks because the man did not explain properly why he was there.
He then tried to flee, which prompted the North Korean troops to shoot ten times at him. When they approached the floating object they only found lots of blood and could not see the man.
The troops decided he was dead and burned the floating object in line with anti-coronavirus rules, according to the North Korean message.
The South Korean government did not know how he came to be in the North, but a defence official said the man might have tried to defect to North Korea. The official said the man was wearing a life jacket on a small floating object and that the military had obtained information that he wanted to go to North Korea.
With additional reporting by the Associated Press