Ukrainian officials said one person was killed and two more injured in Kyiv, with six more, including two children, killed in the regions of Dnipropetrovsk and Poltava.
Klitschko later said several non-residential sites caught fire following the strikes and that a missile struck the home of a large family, warning that the death toll could rise as rescue crews searched through the rubble.
"A dark night," he wrote on Telegram.
The armed forces of Poland, a member of the European Union and NATO, said it scrambled fighter jets to protect its airspace following nearby Russian attacks.
The attack came just hours after Zelenskyy warned that Russia appeared to be preparing a major overnight assault.
"I received a report from Air Force Commander Anatolii Kryvonozhko," Zelenskyy wrote on X Wednesday, saying Russia had prepared a "massive attack several days ago" and that there was a "high probability" it would be carried out overnight.
"It is important that our partners fully understand what is happening and that the protection of lives directly depends on their readiness, or lack thereof, to help us with air defense missiles," Zelenskyy said.
The Ukrainian president made the appeal after returning from the United States, where he met with President Donald Trump.
Trump recently announced at the NATO summit in Turkey that the United States would license Ukraine to manufacture Patriot air defense systems, according to The Associated Press.
"First and foremost, this concerns the United States and those partners that have Patriot missiles and can provide them to support our defense," Zelenskyy wrote.
"Every partner knows what is needed and fully understands the consequences of marking time."
Fox News Digital's Emma Bussey and Reuters contributed to this report.