Among the victims were two parents with their three young boys who "burned alive" in their house, regional deputy police chief Serhiy Blovinov said.
"One whole street... turned into a hellish melted mass," he told Ukrainian TV after the Friday night attack.
Kharkiv Mayor Ihor Terekhov said 15 private houses had burned completely.
President Volodymyr Zelensky vowed a "just response" to Russian "terror".
He identified the three killed children as Oleksiy, 7, Mykhailo, 4 and Pavlo, who was only seven months old.
An elderly couple died in another house that was set ablaze on the same Kotelna street in the city's eastern Nemyshlianskyi district.
Mayor Terekhov said "the Russian aggressor is ruining our city" in a video message from the scene. "But we will overcome. We will win," he added.
He said the Russian attack left 57 local residents Russia left "without a home, without documents, without money, without personal belongings".
Ukraine's officials had earlier said that a petrol station was hit - but later corrected their report.
Kharkiv - Ukraine's second-largest city near the Russian border - has seen almost daily deadly Russian attacks in recent days.
Separately, one man was reported injured in Ukraine's southern city of Odesa in another Russian drone attack on Friday night.
Russia's military has made no public comments on the reported strikes.
Russian President Vladimir Putin launched a full-scale invasion against Ukraine in February 2022.
By Jaroslav Lukiv