Bolot Sher, the acting interior minister, said Sanjar, 25, put up "serious resistance" before being arrested in a special operation.
Kyrgyz authorities said they have "convincing and irrefutable evidence" that Sanjar played a key role in organising the recent ethnic unrest.
Interim authorities have accused his uncle, who was ousted in a violent revolt in April, of hiring "provocateurs" to foment clashes that killed hundreds and displaced more than 100,000.
Bakiyev, who is living in exile in Belarus, has denied any involvement in the violence.
The clashes were the worst ethnic violence to hit the impoverished and strategically important, central Asian country since it gained independence from the Soviet Union nearly two decades ago.
Meanwhile, Kyrgyz soldiers began early voting on Friday in the first stage of referendum to create a new constitution.
"The boys are voting today so they can be on high alert on election day," Abdykalyk Boltabayev, a local election commission official, said.
If successful, the referendum, which begins for the broader population on Sunday, could lead to the creation of central Asia's first democracy.