Zelenskiy pushed the Security Council to expel Moscow from the United Nations and to create a tribunal to investigate actions of the Russian military in Ukraine, which invaded on February 24.
However, Russia is a council veto power and can shield itself from any such action.
"Russia does not have the right to take part in discussing and voting in regard to the war in Ukraine, which is unprovoked and simply colonialist of the part of Russia," Zelenskiy told the council on Tuesday.
"I urge you to deprive the delegation of the terrorist state of its powers."
Moscow calls its invasion a "special military operation" to rid Ukraine of fascists, an assertion the Kyiv government and Western allies say is a baseless pretext for an unprovoked war.
All 15 Security Council members, including Russia, stood for a moment of silence after Zelenskiy asked them to "commemorate all the Ukrainians who have been killed in this war".
Russia's deputy UN ambassador Dmitry Polyanskiy told the council that Zelenskiy's appearance via video had undermined the authority of the body.
"The UN Security Council should not be turned into a platform for a remote PR (public relations) campaign for President Zelenskiy in order to get more weapons from participants of the NATO Summit," Polyanskiy said.
The council met after Ukraine said a Russian missile strike deliberately targeted a shopping centre on Monday, killing at least 18 people.
Russia denied hitting the mall with missiles, saying it had struck a nearby depot of US and European arms.
"We have run out of words to describe the senselessness, futility and cruelty of this war," UN political affairs chief Rosemary DiCarlo told the council, urging all parties to protect civilians and civilian infrastructure.
