A competing text by South Africa, which never mentioned Russia by name, received 50 votes for, 67 against and 36 abstentions, and was therefore not adopted.
The approved resolution names Moscow and "demands an immediate cessation of the hostilities by the Russian Federation against Ukraine, in particular of any attacks against civilians and civilian objects."
The same five countries voted against the resolution Thursday and 2 March: Russia, Syria, North Korea, Belarus and Eritrea.
The US ambassador to the United Nations, Linda Thomas-Greenfield, welcomed the "strong majority" supporting the resolution.
The countries made clear that "Russia bears sole responsibility for the grave humanitarian crisis and violence in Ukraine," she said.
"As president Biden has stated clearly, (Russia's president) Vladimir Putin will not see victory in Ukraine. And we heard today that he will not see it here in New York either."
France's UN envoy Nicolas de Riviere, speaking on behalf of the European Union, praised the resolution as a "very strong signal" sent by the General Assembly which "recalls the urgency of putting an end to the humanitarian catastrophe" and which asks Russia to halt its aggression without delay.
President Joe Biden and his Western counterparts gathered Thursday in Brussels for NATO, G7 and EU summits, with Ukraine president Volodymyr Zelensky also participating by videolink.
On Wednesday Russia submitted a resolution to the UN Security Council on the "humanitarian situation" in Ukraine, but it was not adopted.
Moscow ally China, which has yet to condemn the invasion, joined Russia in voting in favor. But the Security Council's 13 other member states pointedly abstained in solidarity over what diplomats said was the text's unacceptable basis.
Earlier this week UN secretary general Antonio Guterres delivered a cutting assessment of the invasion, calling the war "unwinnable" for Russia.