Russia launched five strikes in Kyiv and against critical infrastructure in the central Dnipropetrovsk and northeast Sumy regions, knocking out electricity to hundreds of towns and villages, said Ukrainian Prime Minister Denys Shmygal.
"Hundreds of settlements were cut off as a result of the attack," he said.
Soldiers fired into the air trying to shoot down the drones after blasts rocked central Kyiv. Residents fled for shelter. An anti-aircraft rocket could be seen streaking into the morning sky, followed by an explosion and orange flames.
Kyrylo Tymoshenko, a senior official at the Ukrainian presidency, said emergency workers rescued 19 people from the rubble, including an elderly woman.
Kyiv mayor Vitali Klitschko said a residential building in the central Shevchenkivsky district of the capital was hit, sparking a fire and damaging several buildings.
Ukraine said the attacks were carried out by Iran-made 'suicide drones', which fly to their target and detonate. Russia's defence ministry said it had carried out a "massive" attack on military targets and energy infrastructure across Ukraine using high-precision weapons.
"Kamikaze drones and missiles are attacking all of Ukraine. The enemy can attack our cities, but it won't be able to break us," President Volodymyr Zelensky said.
"Those who give orders to attack critical infrastructure, to freeze civilians and organize total mobilisation to cover the frontline with corpses, cannot sit at the same table with leaders of (the) G20," senior presidential aide Mykhaylo Podolyak said.
In a statement on social media, he called for Russia to be "expelled from all platforms."
The new United Nations rights chief voiced alarm at the escalating conflict in Ukraine as he began his posting on Monday, insisting that civilians must be protected.
"Any escalation in warfare is deeply troubling to us, and it's happening in Ukraine," UN High Commissioner for Human Rights Volker Turk told reporters in Geneva on his first day in the job.
This report contains information from Reuters and Agence France-Presse