Beside the three who were killed in the attack, at least 35 people are injured and several may still be trapped in the rubble, according to Interior Minister Ihor Klymenko.
"The remains of a 15-year-old boy were found in the rubble," Governor Oleg Synegubov updated the situation on Thursday.
The death of a 12-year-old boy had previously been confirmed, as well as that of a man.
Several apartments in the nine-story residential building were completely destroyed in the attack late on Wednesday.
The attack was carried out with a precision-guided aerial bomb of the type often used by the Russian military in attacks on targets in Ukraine.
Three miles
Ukraine's second-largest city, Kharkiv, which is located about three miles from the Russian border, has been subjected to repeated and extensive Russian air raids during the full-scale Russian war of aggression.
Ukraine's President Volodymyr Zelensky condemns the attack and appeals to partner countries not to delay important decisions.
Appeals for weapons
Ukraine has previously appealed to be allowed to use donated long-range weapons from other countries against targets deep inside Russia to defend itself. Such targets include, among other things, airbases from which Russian bomber planes take off.
Russia has increased pressure in the Kharkiv region in recent months. In the eastern parts of the region, Russia announced on Wednesday that its forces had captured the village of Kruhljavkivka, which is located about two miles from the city of Kupjansk – an important Ukrainian hub that Russia has long tried to encircle.
