Spokesman Andrej Mahecic said the United Nations refugee agency (UNHCR) was advocating with the Bangladesh authorities "to urgently admit these refugees fleeing violence and increasingly hard conditions back home".
An estimated 10,000 to 15,000 have entered Bangladesh through the Anjuman Para border crossing point since Sunday night, many of whom described walking for about a week to reach the border, he said.
"Every minute counts given the fragile condition they're arriving in", said Mahecic.
The delay was due to screening by Bangladesh border guards, he said, emphasising this was the right of any government.
Mahecic said many had chosen to remain in their homes in Myanmar's Rakhine state despite repeated threats to leave or be killed. "They started firing on the village".
Almost 60 per cent of the 582,000 refugees who have fled Myanmar since August 25 are children - and thousands more are crossing each week, UNICEF spokeswoman Marixie Mercado said. "So we escaped into another village", said Mohammad Shoaib, 29.
Kabir said that four other people were injured in the attack and many others fled to safety.
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That figure does not include the thousands now in no man's land.
No fewer than 180 Rohingya have drowned while trying to cross the border over the Naf River since the violence began on August 25.
Thousands of new Rohingya Muslims have arrived in Bangladesh on Monday after fleeing severe hunger in Myanmar.
Al Jazeera's Mohammed Jamjoom reports from the Kutupalong Camp.
A boat carrying refugees along the Naf River sank at dawn on Monday, en route to camps in Cox's Bazar in Bangladesh, killing at least 12 and leaving 35 missing.
The camp would be the largest in the world and has raised concerns about the risks of heavily concentrating such a large number of vulnerable people, such as the spread of disease.