Strikes on key facilities targeted “the full Ukrainian nation” and were an effort to force Kyiv to surrender, the prosecutor-general told the BBC.
The term genocide refers to an effort to wipe out a group of people. Russia denies having any such aim.
Millions of people across Ukraine are facing power cuts in freezing weather, following sustained Russian attacks.
Efforts are ongoing to reconnect homes cut off from electricity. Officials say Kherson has now been fully resupplied, following the city’s recapture by Ukrainian troops earlier this month.
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But people in 14 regions and the capital Kyiv remain under usage restrictions, according to President Volodymyr Zelensky.
Genocide involves “the intent to destroy, in whole or in part, a national, ethnic, racial or religious group”, according to a definition provided by the UN Genocide Convention.
Among the acts which may qualify are killing or causing serious harm to members of that group – or forcibly transferring its children elsewhere.
Mr Kostin said his office was investigating reports of more than 49,000 war crimes and crimes of aggression since Russia launched its full-scale invasion on 24 February.
The “same pattern of conduct” was being seen in every Ukrainian settlement occupied by Russian forces, he observed.
A war crime constitutes a violation of the so-called “rules” of war set out by international treaties including the Geneva Conventions.
Among other things, the Conventions stipulate that civilians must be protected. Russia has been repeatedly accused of breaking this rule.
Shelling of residential buildings in Dnipro over the weekend left one person dead and 13 others injured, according to the governor of the wider region.
Meanwhile, police said 32 civilians had been killed by strikes in Kherson since Russian troops left the southern city earlier this month.
On Saturday, Ukraine marked the 90th anniversary of the Holodomor – a man-made famine that killed millions of people during Soviet rule.
Of the war crimes seen since Russia invaded in February, Mr Kostin said 260 people had been indicted and 13 verdicts had been issued by Ukrainian courts.
He called for the creation of an “international ad-hoc tribunal”, supported by countries of “the whole civilised world” who opposed the invasion to hold Russia to account.