Members of 38 families said they fled their homes fearing attack from military helicopters, said Colonel Zahirul Haque Khan, the Border Guards Bangladesh (BGB) commander in Bandarban district where the group of 136 people are now living in shelters.
Clashes between Myanmar's military and the insurgent Arakan Army, which mainly recruits from the Rakhine ethnic group, have displaced more than 5,000 people in parts of Rakhine and Chin states since early December.
While Rakhine State is majority Buddhist, in 2017 attacks on security posts by insurgents from the Muslim Rohingya minority provoked a military crackdown that forced 730,000 people from their homes and into camps in Bangladesh's Cox's Bazar district, according to UN agencies.
Win Thein, a member of the nongovernmental Bangladesh Human Rights Commission, said he had visited the refugees, who are members of the Khumi, Cho and Rakhine ethnic groups, in their remote jungle camp.
They had crossed into Bangladesh on Sunday and Monday after fleeing from two villages in Chin state's rugged Paletwa township after they heard gunfire and saw helicopters nearby, he said.
Some villagers said they later witnessed Myanmar soldiers looting and setting fire to homes, he said.
Win Thein said some of the refugee children were seriously ill and had no access to medical care.
"There are no blankets at all and it is very cold," he said.
Bangladesh summoned Myanmar's ambassador on Tuesday to protest over the new arrivals, according to a senior Bangladesh foreign ministry official.
Brigadier General Sazedur Rahman, a BGB regional commander, said Bangladesh had beefed up security near the border to prevent more refugee arrivals.
The United Nations refugee agency in Cox's Bazar was trying to gather more information about the new arrivals, said spokesman Firas Al-Khateeb.
"We'll coordinate with the government so we will be able to help them as soon as possible," he said.