Mr Zelensky was speaking at a UK-convened meeting of the UN’s most powerful body in New York, days after the discovery of hundreds of murdered civilians in the Ukrainian town of Bucha.
Russia has a veto on all UN resolutions as one of the five permanent members of the Security Council, known as the P5. It recently blocked one calling for an immediate end to the invasion of Ukraine.
The UN can send peacekeepers to warzones and back military action by its members if the P5 agree. However, the international body is racked by growing rivalry between the West, Russia and China.
The divides were still apparent on Tuesday, with Britain and the US condemning Russia’s atrocities, while China and the United Arab Emirates – a non-permanent member of the council – avoided any criticism of Moscow.
In his address, Mr Zelensky lambasted the UN for failing to prevent massacres in Syria, Somalia, Afghanistan and Libya. If “tyranny” had been countered earlier by the UN, then his country would not now be at war, he said.
He proposed convening a global conference in Kyiv to reform the world security system, adding: “It is time to transform the system of the United Nations.”
Mr Zelensky said any Russian military leader or politician who had given an order that led to the carnage in his nation should be brought before a court “similar to the Nuremberg tribunals”.
He said: “I would like to remind Russian diplomats that Von Ribbentrop did not escape punishment for crimes after the Second World War. Also, I remind you that Adolf Eichmann did not go unpunished.”
Mr Zelensky showed the 15-member Security Council a graphic 90-second video of the aftermath of Russian atrocities, including dead women and children, burned bodies and victims in mass graves.
He said: “Russian troops are deliberately destroying Ukrainian cities to ashes with artillery and airstrikes. They are deliberately blocking cities, creating mass starvation. They deliberately shoot columns of civilians on the road trying to escape from the hostilities. They even deliberately blow up shelters where civilians hide from airstrikes.”
The Russians had deported hundreds of thousands of Ukrainians and abducted more than 2,000 children, he said.
Mr Zelensky added: “Russia wants to turn Ukraine into silent slaves.”
Grim evidence continued to emerge of massacres by Russian forces on the outskirts of Kyiv.
Ukrainian officials said the bodies of more than 400 civilians had been found in towns around the capital.