The General Assembly decided, with 107 votes in favour, that it would hold a public vote - and not a secret ballot - on a draft resolution that would condemn Russia's "illegal so-called referenda" and the "attempted illegal annexation."
Diplomats said the vote on the resolution would likely be on Wednesday.
Only 13 countries on Monday opposed holding a public vote on the draft resolution, another 39 countries abstained and the remaining countries did not vote.
Russia had argued that a secret ballot was needed because Western lobbying meant that "it may be very difficult if positions are expressed publicly."
Moscow has moved to annex four partially occupied regions in Ukraine - Donetsk, Luhansk, Kherson and Zaporizhzhia - after staging what it called referendums. Ukraine and allies have denounced the votes as illegal and coercive.
The draft resolution to be voted on later this week calls on states not to recognise Russia's move and reaffirms the sovereignty and territorial integrity of Ukraine.
US Secretary of State Antony Blinken pressed the international community earlier on Monday to make clear that Russian President Vladimir Putin's actions are "completely unacceptable."
"Now is the time to speak out in support for Ukraine; it is not the time for abstentions, placating words, or equivocations under claims of neutrality. The core principles of the UN Charter are at stake," Blinken said in a statement.
Russia vetoed a similar resolution in the 15-member Security Council last month. It has been trying to chip away at its international isolation after nearly three-quarters of the General Assembly reprimanded Moscow and demanded it withdraw its troops within a week of its February 24 invasion of Ukraine.
The moves at the United Nations mirror what happened in 2014 after Russia annexed Ukraine's Crimea. At the Security Council Russia vetoed a draft resolution that opposed a referendum on the status of Crimea and urged countries not to recognize it.
The General Assembly then adopted a resolution declaring the referendum invalid with 100 votes in favour, 11 against and 58 formal abstentions, while two dozen countries didn't take part.
Call to designate Russia as terrorist state
Hundreds of people protested outside the Russian embassy in Warsaw on Monday, calling for Russia to be designated a terrorist state and for its diplomats to be expelled after missile strikes killed at least 11 people in Ukrainian cities on Monday.
Opposition to the Russian invasion is intense in Poland, with the country becoming one of Kyiv's staunchest supporters and taking in millions of refugees since the conflict started.
"All my family ... stayed in Ukraine and I don't know how I can support them," said Valeria Horna, a student from Kyiv who lives in Warsaw. "That's why I came to this meeting, simply to support Ukraine."
Protestors, many draped in the Ukrainian flag, brandished placards with the protest's slogan "Russia is a terrorist state."
"I demand that Russia be designated a terrorist state because... it is bombarding the innocent civilian population," said Monika Lowicka, a 40-year-old tax adviser.
A representative of the organisers, Euromaidan-Warszawa, put the number of protestors at around 2000. A police spokesman declined to comment on how many people were present.
Russia calls its actions in Ukraine a “special operation” that it says is not designed to occupy territory but to destroy its southern neighbour's military capabilities and capture what it regards as dangerous nationalists.
