Refugees who survived the perilous journey said they were driven out by hunger because food markets in Myanmar's western Rakhine State have been shut and aid deliveries restricted. They also reported attacks by the military and Rakhine Buddhist mobs.
The influx will worsen the unprecedented humanitarian emergency unfolding in Cox's Bazar, where aid workers are battling to provide refugees with food, clean water and shelter.
On Monday, the Red Cross opened a field hospital as big as two football fields, with 60 beds, three wards, an operating theatre, a delivery suite with maternity ward and a psychosocial support unit.
Hundreds of thousands of Rohingya had already been in Bangladesh after fleeing previous spasms of violence in Myanmar, where they have long been denied citizenship and faced curbs on their movements and access to basic services.
The United States and the European Union are considering targeted sanctions against Myanmar's military leaders, officials have told Reuters.
EU foreign ministers will discuss Myanmar on Monday, and their draft joint statement said the bloc "will suspend invitations to the commander-in-chief of the Myanmar/Burma armed forces and other senior military officers".
The powerful army chief, Min Aung Hlaing, told the United States ambassador in Myanmar last week that the exodus of Rohingya, whom he called non-native "Bengalis", was exaggerated.
But despite Myanmar's denials and assurances that aid was on its way to the north of violence-torn Rakhine State, thousands more starving people were desperate to leave.
"We fled from our home because we had nothing to eat in my village," said Jarhni Ahlong, a 28-year-old Rohingya man from the southern region of Buthidaung, who had been stranded on the Myanmar side of the Naf for a week, waiting to cross.
From the thousands gathered there awaiting an opportunity to escape, about 400 paid roughly $50 each to flee on nine or 10 boats on Monday morning, he added.
"I think if we go to Bangladesh we can get food," he said.
