Thousands of people are thought to be trapped in an unoccupied zone between the countries.
Independent confirmation of the figures is almost impossible as few journalists are given access to the volatile region.
Aid workers giving emergency shelter and food in Bangladesh said the refugees were “in a very, very desperate condition”.
“People are traumatised, which is quite visible,” said Sanjukta Sahany of the International Organization for Migration (IOM).
About a dozen of the new arrivals had recent bullet wounds, said aid workers
What is happening in Rakhine?
Rakhine, the poorest region in Myanmar (also called Burma), is home to more than a million Rohingya.
The Rohingya have faced decades of persecution in Buddhist-majority Myanmar, where they are not considered citizens. There have been sporadic waves of deadly violence in recent years.
The current escalation is the most significant outbreak of violence in Rakhine since October 2016, when nine policemen died in similar attacks on border posts.
The violence comes just days after an international commission led by former UN chief Kofi Annan warned of more radicalisation if ethnic tensions were not addressed.
Refugees arriving in Bangladesh accuse the military in Myanmar of raiding villages and burning down houses.
With media access to Rakhine severely restricted, casualties and violence are difficult to verify.
An AFP journalist on a government-led trip to the province said he saw plumes of smoke rising from several burning villages.
The government said that mine explosions and fighting continued, but blamed the militants for burning down houses and afterwards fleeing to the mountains.
What is it like for the refugees?
By Mir Sabbir, BBC News, Cox’s Bazar in Bangladesh
Along the main road in front of Kutupalong refugee camp, hundreds of Rohingya gather in small groups under the open sky. The vast majority are women and children.
These people crossed the border under the cover of darkness, walked a long way or took an auto-rickshaw if they could afford one, and reached this camp. They know if they are caught during the crossing, they will be sent back.
Image captionThis woman had to wait at the border for three days
But even after being sent back, they just keep trying again. Some say there is nothing left for them at home. Many of them have lost their family members.
Many of the Rohingya crossing the border already have relatives in these camps. With their help, these people will probably also get some shelter here. But looking at children in the refugee camps – exhausted by the long journey, confused about why they had to come here and suddenly living under the open sky – it’s hard not to get emotional.
Who are the militants?
A group called the Arakan Rohingya Salvation Army (Arsa) has said it carried out the attacks on police stations.
The group first emerged in October 2016, when it carried out similar assaults on police posts, killing nine police officers.
Those attacks triggered a military crackdown that led to widespread allegations of killings, rape and torture of Rohingya, and an exodus of Rohingya into Bangladesh.
Image captionTens of thousands of Rohingya Muslims have fled to Bangladesh