The Rohingya villagers who carried the victims' bodies said they had been killed by the Myanmar military on Wednesday, Bangladesh border guard, commander Manzurul Hasan Khan, told AFP.
They were buried in a Muslim graveyard along with a sixth man who died of unknown causes.
"Five of them had bullet wounds," Chailau Marma, deputy police chief of Cox's Bazar, which borders Myanmar, told AFP.
"Relatives of the deceased carried the bodies into Bangladesh after they failed to bury them in Myanmar. They all died yesterday."
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Last week border guards had allowed a Rohingya couple to be buried in Bangladesh after they were shot dead and the wife gang-raped when they headed back to their Myanmar village, Khan said.
Rohingya refugees have poured into Bangladesh, fleeing a massive security sweep in Rakhine state by Myanmar forces following a series of deadly ambushes there by Rohingya militants on August 25.
The United Nations says 164,000 refugees have so far entered Bangladesh.
Around 50 gunshot victims have been taken to a major hospital in the city of Chittagong, some 200 kilometres (125 miles) north of the Myanmar border because the local clinics lack the facilities to treat their injuries.
Police said two of those taken to the hospital have died.