President Volodymyr Zelenskiy said the strike late on Wednesday night (local time) - the latest in a long series of attacks on the city - underscored the need for more help from Ukraine's Western backers.
He pointed to Iran's strike on Israel as an example of allies working together.
In order to stop Russian strikes, "Ukraine must receive the necessary, and most importantly, sufficient help from the world, from our partners.
"Every leader knows exactly what needs to be done. It's important to be decisive," Zelenskiy said in a posting on the Telegram messaging app.
Kharkiv regional governor Oleh Syniehubov said the bomb hit between the third and fourth floors of the building in the city's Saltivka district.
"Several floors have been destroyed. An apartment by apartment search is under way. People could be under the rubble," Syniehubov said in a video posted online.
Pictures posted online showed cars ablaze outside the apartment block and firefighters making their way through smoke rubble to get inside the building.
Kharkiv Mayor Ihor Terekhov put the injury toll at 10, including a three-year-old child. He said guided bombs had struck two city districts.
Located 30km from the Russian border, Kharkiv has been a frequent target of Russian forces throughout the more than two and-a-half-year war.
In Kyiv, the head of the capital's military administration said fragments from a downed Russian drone damaged an apartment building in one of the capital's eastern districts. There was no indication of any casualties.
Russia denies targeting civilians, but has regularly struck towns and cities behind the front line.
In his nightly video address, Zelenskiy referred to the help the United States and other partners provide Israel to fend off attacks.
"Every time in the Middle East, during criminal Iranian strikes, we see how the international coalition acts together," he said, echoing comments he made during an April raid launched by Iran on Israel.
Zelenskiy also issued the latest of a series of calls for more help to be agreed at a meeting this month in Germany devoted to providing Ukraine with military assistance.
US President Joe Biden is to attend the meeting.
