The incident came after a Rohingya woman had a leg blown off in the same area on Monday, raising fears that the border area had been deliberately mined.
"They stepped onto some sort of explosives this morning and one of them lost his leg," border guard commander Manzurul Hasan Khan told AFP on Tuesday of the two children.
It is not known what caused the blast, which he said was well inside Myanmar territory, but Khan said he believed it was a landmine.
Khan said a Rohingya woman had been brought to the border on Monday after losing half her leg in a blast, hours after guards heard a loud explosion from the Myanmar side.
Khan said many Rohingya were also entering Bangladesh with bullet wounds, although it was impossible to say how these were sustained as media access to the worst-hit parts of Myanmar's neighbouring Rakhine state is limited.
All three have been taken to hospitals in Cox's Bazar, the nearest city to the border, where hundreds of thousands of Rohingya -- a stateless Muslim minority in mainly Buddhist Myanmar -- have taken shelter in camps.
The victims were among around 90,000 Rohingya who have abandoned their homes in Rakhine for Bangladesh since a fresh outbreak of fighting in the state on August 25.
Their arrival has raised fears of a fresh humanitarian disaster as already crowded camps in Bangladesh struggle to cope with the influx.
The latest unrest broke out when a Rohingya militant group launched a series of coordinated ambushes on Myanmar security posts in response to what it said was a fresh crackdown.
The Rohingya are seen as illegal immigrants in Myanmar and have suffered decades of persecution, according to rights groups.
Unverifiable testimony from those who have fled indicates tit-for-tat mass killings and villages being torched by the army, Buddhist mobs and Rohingya militants.
In addition to the Rohingya at least 11,000 Rakhine Buddhists and Hindus have fled arson and attacks by militants to camps inside Myanmar, according to the last government update.
The UN said on Monday that 87,000 mostly Rohingya refugees had arrived in Bangladesh since the latest eruption of violence and around 20,000 more were massed on the border waiting to cross.
Ali Hossain, the government administrator of the Cox’s Bazar district, said the number of refugees would likely pass 100,000 on Tuesday.
"We’ve estimated that until yesterday, some 92,000 Rohingya have arrived since August 25. By today, the number could soar past 100,000," he said.
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