Regional head Oleh Syniehubov said one of the boys, aged 12, was fatally injured when the building was hit by a Russian 500-kilogram (1,100-pound) glide bomb.
"He was freed from under the rubble with severe head injuries and fractures," Syniehubov wrote on social media. "Doctors performed resuscitation measures for more than half an hour. Unfortunately, it was not possible to save the child."
Syniehubov said later that rescuers also retrieved the bodies of a 15-year-old boy and an unidentified man from the debris.
Interior Minister Ihor Klymenko said at least 35 people were injured in the attack and others could still be trapped under the rubble.
Russia has increasingly used powerful glide bombs to pummel Ukrainian positions along the 1,000-kilometer (600-miles) line of contact and strike cities dozens of kilometers (miles) from the front line. Kharkiv, a city of 1.1 million, is about 30 kilometers (less than 20 miles) from the border.
President Volodymyr Zelenskyy has repeatedly urged the United States to allow Ukraine to use long-range American missiles to strike air bases deeper in Russia that are used by warplanes carrying glide bombs. Washington so far has only allowed some strikes close to the border.
Zelenskyy repeated his request Thursday, publishing a video showing the ravaged nine-story building, at least three of its floors destroyed and the rest of it seriously damaged.
"Partners see what is happening every day," Zelenskyy wrote on Telegram. "And under these conditions, each of their delayed decisions means at least dozens, if not hundreds of such Russian bombs against Ukraine. Their decisions are the lives of our people. Therefore, we must together stop Russia and do it with all possible force."
Early Thursday, Russia also fired 10 missiles of various types at the Dniester Estuary bridge that connects the northern and southern parts of the Odesa region, Ukraine's air force said, adding that only two of them were intercepted. It didn't say whether the bridge was hit.
Russian forces also sent 43 exploding drones over at least nine Ukrainian regions, the air force said. It said 17 were shot down, 23 jammed and three flew back to Russian-controlled territories.
The head of Kyiv city administration, Serhii Popko, said debris from some of the drones that were intercepted over the capital fell in the Podil district just north of downtown, causing minor damage. According to Popko, there was only one day in October in which Kyiv residents did not hear an air alert. During the month, Kyiv experienced two air alerts per day on average, repelling 20 actual aerial attacks targeting the city.
In the country's east, Russian forces continued their slow but steady assault, trying to capitalize on Ukrainian shortages of manpower and ammunition. In Moscow, the Defense Ministry reported the capture of the village of Yasna Polyana in the Donetsk region that lies on the way to the well-fortified Ukrainian stronghold of Kurakhove.
Ukraine struck back Thursday with drone attacks.
Authorities in the Russian-occupied city of Berdyansk on the Sea of Azov reported a drone attack on the port early Thursday, saying three people were injured.
Russia's Defense Ministry said air defenses on Thursday downed 21 Ukrainian drones over several regions and the Black Sea.
Zelenskyy, who has rallied support of Ukraine's western allies for his "victory plan," said Thursday that the country has also worked to prepare a bilateral document with Hungary aimed at "avoiding challenges" between Kyiv and Budapest.
Zelenskyy, who made the statement while speaking to the heads of territorial communities and districts of Ukraine's westernmost Zakarpattia region that borders Hungary, said the prospective document will address the matters of Ukraine's security and its bid to join NATO.
"The wording is very gentle, which may allow us to start a dialogue with Hungary," Zelenskyy said, noting that Ukraine would ask Hungary "not to block Ukraine's invitation to NATO. We're not asking to support us or to vote for us, just not to block," Zelenskyy added.
Prime Minister Viktor Orbán, Russian President Vladimir Putin's closest partner within the European Union, has played the spoiler on showing unity in confronting Russia over its war in Ukraine and spoken strongly against issuing an invitation for Ukraine to join NATO.