Tamerlan Tsarekayev, a 19-year-old military academy student, died on Wednesday morning, health ministry officials told Interfax news agency.
A woman passerby had died on the spot and eight others were injured when a car exploded in the centre of the regional hub shortly after 14:00 GMT on Tuesday.
The car blew up as a truck filled with soldiers drove by.
Tsarekayev was initially reported dead, but authorities said early on Wednesday that he was in critical condition at a local hospital after undergoing surgery. He died shortly after this statement.
Three other soldiers are in hospital in a stable condition and the rest of the injured received minor injuries, Interfax reported.
Authorities have not yet determined what caused the blast, a car bomb or a gas canister, and did not immediately link the explosion to attacks by Chechen rebels, which have been staged in Russia over the past several years.
Interior ministry officials of North Ossetia, where Vladikavkaz is located, said the blast could have been directed at the Gamid Bank, in front of which it occurred.
The last attack blamed on Chechen rebels killed six people and the female bomber outside the National Hotel in downtown Moscow, a stone's throw from the Kremlin.
In March 1999, a bomb blast at a Vladikavkaz market killed 55 people and injured more than 150 others.
Vladikavkaz, the capital of the North Ossetia republic, lies about 30km west of war-ravaged Chechnya, where fighting between Russian troops and separatist rebels has simmered for more than four years.
Russian President Vladimir Putin sent Russian troops into the restive republic in October 1999 in what the Kremlin said was an anti-terrorist operation.