13-year-old Prince Alexei would have been the heir to the throne
DNA tests will be carried out on the bones, thought to be those of Prince Alexei and his elder sister Maria.
Archaeologists excavated ground close to the site in Yekaterinburg where the tsar, his wife and their three other daughters were found in 1991.
The prosecutor-general is reopening an investigation into the case.
Archaeologist Sergei Pogorelov says bullets found at the burial site indicate the children had been shot.
He told Russian television the newly unearthed bones belonged to two young people: a young male aged roughly 10-13 and a young woman about 18-23.
Ceramic vessels found nearby appear to have contained sulphuric acid, consistent with an account by one of the Bolshevik firing squad, who said that after shooting the family they doused the bodies in acid to destroy the flesh and prevent them becoming objects of veneration.
The regional forensic bureau chief, Nikolai Nevolin, told Itar-Tass news agency that 44 bone fragments had been handed over, and would be subjected to detailed analysis.
"We know that Prince Alexei suffered from haemophilia, so we'll have to detect the genome of this disease," he said.
The bullets found at the burial site would also be tested, he said.
Nicholas abdicated in 1917, and he and his family were detained. The following year, they were sent to Yekaterinburg in the Ural Mountains, where a Bolshevik firing squad executed them on July 17, 1918.In 1998, experts exhumed and ceremonially reburied what were widely considered to be the remains of Nicholas, Alexandra, and the three daughters.