Five of the dead were in the Kyiv region outside the capital, where Ukraine's President Volodymyr Zelenskiy said Russian forces targeted energy infrastructure, but also damaged residential buildings, schools and businesses.
Fifteen people were injured in Saturday's attack.
Zelenskiy said the Sumy, Kharkiv, Dnipro and Mykolaiv regions were also targeted in an attack that included about 430 drones and 68 missiles, most of which were downed by air defences.
Later in the day, the governor of the southeastern region of Zaporizhzhia, Ivan Fedorov, said a residential area of the city of Zaporizhzhia had been hit by Russian-guided bombs, killing one person and injuring three.
Reuters Television footage showed emergency crews at work amid piles of rubble and twisted metal. Windows and frames on balconies were smashed.
"The second attack was strong. The kitchen window flew out, as did those in the living room and bedroom, everything went into a roar. I ran and got slightly injured," Olha Kiyashko, 65, told Reuters.
"I've got no strength left. They took away our pension years - the years we could have lived on, the years we'd planned and counted on. All our health is gone."
In Moscow, Mayor Sergei Sobyanin said Russian air defence units downed 65 Ukrainian drones headed for the city throughout Saturday.
Sobyanin, writing on Telegram, said the drones were intercepted over an eight-hour period.
The governor of Bryansk region on the Ukrainian border, Alexander Bogomaz, said on Telegram that units in his region had downed 128 drones.
Ukrainian forces earlier this week said they had struck a key plant producing missile components in Bryansk region. Bogomaz said seven people died in that attack, without saying what was hit.
According to the most recent report issued by the Russian Defence Ministry, air defence units intercepted 280 Ukrainian drones on Saturday in various parts of central and western Russia over a 10-hour period on Saturday.
Saturday's strikes come as the Iran conflict has distracted international attention from a US-backed peace push in the four-year war, which Kyiv says Moscow has no interest in ending.
"Russia will try to exploit the war in the Middle East to cause even greater destruction here in Europe, in Ukraine," Zelenskiy wrote on social media platform X.
He repeated his call for Kyiv's partners to boost production of air-defence weapons, stocks of which have been diminishing as the US and its allies in the Gulf have fended off Iranian strikes.
Russia's winter attacks on Ukraine have left swathes of major cities without power or heating, part of a campaign to weaken resolve as Moscow's troops press a battlefield offensive and demand Kyiv cede more territory in the east.
Ukraine's forces have targeted Russian strategic infrastructure such as oil refineries, depots and terminals in long-range strikes.
Ukraine's Energy Ministry said on Saturday that consumers in six regions were without electricity after the overnight strikes and Russian shelling of frontline areas.
Saturday's attack also prompted NATO member Poland to scramble jets to protect its airspace, but no violations were observed, Warsaw's military said.
In Moldova, on Ukraine's western border, the Foreign Ministry denounced what it said was an intrusion by a Russian drone into its airspace in a border district, saying Moscow's actions undermined regional security and posed a danger to its citizens.
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