Russian forces used guided bombs in the attack, news outlets reported.
Kharkiv Mayor Ihor Terekhov said at least four explosions were heard in a densely populated residential area in the afternoon, leading to civilian deaths and injuries.
Prosecutors put the casualty toll at three dead and 52 injured in the attack, including three injured children.
Regional Governor Oleh Syniehubov said four of those hurt were in serious condition.
Rescue teams were working on the site of the attack, Ukrainian state news agency Ukrinform reported.
'Bold decisions from our partners are needed': Zelensky
President Volodymyr Zelensky called for more Western help to deal with the growing threat of guided bombs.
"Russian terrorists have struck Kharkiv with guided aerial bombs again. Unfortunately, a residential building was hit," Zelensky wrote on social media.
He added: "This Russian terror with guided aerial bombs must and can be stopped. Bold decisions from our partners are needed so that we can destroy Russian terrorists and Russian combat aircraft where they are."
"We have already proved that it is possible to protect people’s lives from missile terror, particularly by clearing the border areas of terrorist launchers," Zelensky also said. "Protection against bombs is required as well. We need this determination. I thank everyone who helps Ukraine protect lives."
Kharkiv, Ukraine's second-largest city, lies about 30 kilometres from the border with Russia.
The city, which is home to around 1.3 million people, has frequently been targeted by Russia since it invaded Ukraine nearly 28 months ago, the Reuters news agency reported.
Russia invaded Ukraine on February 24, 2022, starting the largest armed conflict in Europe since World War II.
Saturday is day 850 of Russia’s war on Ukraine.
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Source: IAR, PAP, Reuters, ukrinform.net