The border authorities came under fire just after midnight and shot back at two fishing trawlers along a river dividing Bangladesh from Myanmar's westernmost state of Rakhine, police said.
"The BGB (Border Guard Bangladesh) later found the boat anchored at a river island. One woman was found shot dead and four were injured," Mainuddin Khan, the police chief in the Bangladeshi border town of Teknaf, told AFP.
Police also discovered 28,000 "yaba" - or methamphetamine tablets - aboard the boat, Khan said.
But the head of Teknaf's government-run hospital, Iskandar Mirza, said the woman identified as 50-year-old Jahida Khatun and the four injured were all Rohingya.
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"Her bullet-ridden body was brought to the hospital, while four injured Rohingya men were brought in," he told AFP.
Some 75,000 Rohingya refugees have entered Bangladesh since October, when government forces in Myanmar unleashed a bloody crackdown on the Muslim minority.
Many recalled horrific stories of villages being torched, relatives burned alive and the gang rape of women by security forces.
The sudden influx in recent months prompted Bangladesh to resurrect a controversial plan to relocate refugees to an undeveloped island in the Bay of Bengal.
The UN rights council has agreed to investigate allegations of rape, murder and torture against the army, though Myanmar has denied its troops have been waging a campaign of ethnic cleansing.
Myanmar denies citizenship to most of the million-strong Rohingya in Rakhine, despite many living there for generations.