The drone attack damaged a residential building and several windows were also blown out at a kindergarten and clinic, said Saratov regional Governor Roman Busargin.
Russia's defence ministry said it had shot down 41 Ukrainian drones over Russian territory overnight.
In Ukraine, Russia launched overnight drone and missile strikes on five Ukrainian regions, targeting energy and port infrastructure. Ukrainian Interior Minister Ihor Klymenko said Saturday that over a million people were without electricity in the country.
Ukrainian President Volodymyr Zelenskiy said Russia had sent over 450 drones and 30 missiles into Ukraine overnight.
An attack on the Black Sea city of Odesa caused grain silos to catch fire at the port, Ukrainian deputy prime minister and reconstruction minister Oleksiy Kuleba said.
Two people were wounded in attacks on the wider Odesa region, according to regional head Oleh Kiper.
Kyiv and its Western allies say Russia is trying to cripple the Ukrainian power grid and deny civilians access to heat, light and running water for a fourth consecutive winter, in what Ukrainian officials call "weaponising" the cold.
The latest round of attacks came after Kremlin adviser Yuri Ushakov said Friday that Russian police and National Guard will stay on in eastern Ukraine's Donbas and oversee the industry-rich region, even if a peace settlement ends Russia's nearly four-year war in Ukraine.
This underscores Moscow's ambition to maintain its presence in Donbas post-war. Ukraine is likely to reject such a stance as US-led negotiations drag on.
Moscow will give its blessing to a ceasefire only after Ukraine's forces have withdrawn from the front line, Ushakov said in comments published in Russian business daily Kommersant.
Meanwhile, Germany is set to host Zelenskiy on Monday for talks as peace efforts gain momentum and European leaders seek to steer negotiations.
For months, American negotiators have tried to navigate the demands of each side as US President Donald Trump presses for a swift end to Russia's war while growing increasingly exasperated by delays.
The search for possible compromises has run into a major obstacle over who keeps Ukrainian territory currently occupied by Russian forces.
