He said people died in their homes and on the streets and were crushed by tanks in their cars "for pleasure".
Russia should be held accountable for what had happened, he said, and removed from the UN Security Council.
Horrific images of bodies in streets in towns such as Bucha have generated shock and condemnation worldwide.
After his speech by video link to the Security Council, Zelensky showed a graphic video of images of dead Ukrainians, some burned and disfigured.
"The Russian military searched for and purposefully killed anyone who served our country," he said.
"They killed entire families, adults and children, and they tried to burn the bodies."
Russian UN Ambassador Vasily Nebenzia said there was no evidence Russian troops had carried out atrocities.
"We've heard once again a huge amount of lies about Russian soldiers and military," he said.
Zelensky said the UN had to act, because not doing so would mean there was no point to the world body.
"Are you ready to close the UN? And the time of international law is gone? If your answer is no, then you need to act immediately," he said, adding that "accountability must be inevitable".
"We are dealing with a state that turns its veto at the UN Security Council into the right to [cause] death," he said.
He called for the establishment of a special international tribunal to deal with the crimes.
There was "not a single crime they would not commit", Zelensky said.
"The world has yet to see they have done in other occupied cities and regions of our country."
UN Secretary-General António Guterres said he would "never forget" seeing images of dead civilians in Bucha.
He warned the Security Council that Russia's invasion is one of the greatest challenges ever to the international order "because of its nature, intensity, and consequences".
The US reiterated its call for Russia to be suspended from the UN Human Rights Council.
Linda Thomas-Greenfield, the US ambassador to the UN, told the Security Council: "Russia should not have a position of authority in a body whose very purpose is to promote respect for human rights."
China, which has avoided criticising the Russian invasion, said that the reports of civilian deaths in Ukraine were "very disturbing", but said any accusations should be based on verified facts.
The Ukrainian government started a war crimes investigation on Monday after it said the bodies of 410 civilians had been found in areas around Kyiv.
The images from Bucha prompted US President Joe Biden to call for his Russian counterpart Vladimir Putin to be tried for war crimes.
A growing number of Western nations have expelled Russian diplomats in response to the discovery of the atrocities, while new sanctions against Russia are being discussed.
European Commission President Ursula von der Leyen and foreign affairs chief Josep Borrell will travel to Kyiv to meet the Ukrainian president later in the week, von der Leyen's spokesperson said.