More than 600,000 Rohingya have arrived in Bangladesh since a military crackdown in Myanmar in August triggered an exodus, straining resources in the impoverished country.
Police said the attack by the Rohingya man was linked to a family dispute. It is the latest of a string of crimes in the area which have alarmed local authorities.
"We've stepped up security after these incidents," deputy chief of Cox's Bazar district Afruzul Haq Tutul told AFP.
The district's state prosecutor Mamtaz Ahmed told AFP that crime had recently increased in the area.
The UN has described Myanmar's crackdown on the Rohingya as "textbook" ethnic cleansing, and many of the refugees who arrive in Bangladesh bring horrific stories of brutalities including murders, rapes and arson.
This has prompted an unprecedented outpouring of sympathy in Muslim majority Bangladesh.
Locals complain that Rohingya are squeezing out poor Cox's Bazar villagers from the job market, especially in the booming construction sector and the fishing industry.
"The majority of the fishermen in our fishing trawlers are Rohingya. They are hard working and they can be employed at a low cost," Jashim Uddin, who owns a small fishing trawler, told AFP.
Others complained the influx of Rohingya had caused food prices and transport costs to spike.
"The vegetable prices are at a record high. But nobody seems to care or listen to our woes. They only care for the Rohingya," said Nazir Ahmed, a Bangladeshi resident of Ukhia.