Ukrainian officials said people were banned from leaving some occupied areas until the four-day vote was over, armed groups were going into homes, and employees were threatened with the sack if they did not participate.
Ukrainian President Volodymyr Zelenskiy said in a nightly address that the votes would be "unequivocally condemned" by the world, along with the mobilisation Russia began this week, including in Crimea and other areas of Ukraine occupied by Russia.
"These are not just crimes against international law and Ukrainian law, these are crimes against specific people, against a nation," Zelenskiy said.
The votes on becoming part of Russia were hastily-organised after Ukraine recaptured large swathes of the northeast in a counter-offensive earlier this month.
With Russian President Vladimir Putin also announcing a military draft this week to enlist 300,000 troops to fight in Ukraine, the Kremlin appears to be trying to regain the upper hand in the grinding conflict since its February 24 invasion.
Zelenskiy also addressed people in parts of Ukraine occupied by Russia, and said they should resist efforts to mobilise them to fight.
"Hide from Russian mobilisation in any way you can. Avoid draft orders. Try to move to the territory of free Ukraine," he said, urging those who did end up in the Russian armed forces to "sabotage," "interfere" and pass on intelligence to Ukraine.
By incorporating the four areas, Moscow could portray attacks to retake them as an attack on Russia itself - potentially using that to justify even a nuclear response.
Putin and other Russian officials have mentioned nuclear weapons as an option in extremis: a terrifying prospect in a war that has already killed tens of thousands of people, uprooted millions and pummelled the global economy.
Voting in the provinces of Luhansk, Donetsk, Kherson and Zaporizhzhia in the east and southeast, representing about 15 per cent of Ukrainian territory, was due to run from Friday to Tuesday.
In the Donetsk region, the turnout on Friday was 23.6 per cent, Tass cited a local official as saying. More than 20.5 per cent of voters eligible to vote in the Zaporizhzhia region and 15 per cent of those in the Kherson region voted on Friday, Russia's Interfax news agency reported, citing local electoral officials.
Polling stations were also set up in Moscow, for residents of those regions now living in Russia. Flag-waving government supporters attended rallies in Moscow and St. Petersburg in favour of the referendums and the war effort.
Ukraine, Western leaders and the United Nations condemned the votes as an illegitimate precursor to illegal annexation. There are no independent observers, and much of the pre-war population has fled.
NATO Secretary-General Jens Stoltenberg said the military alliance would step up support for Ukraine in response to the referendums.
"We will never recognise these referenda which appear to be a step toward Russian annexation and we will never recognise purported annexation if it occurs," added the Group of Seven leading industrial democracies.
Moscow says they offer an opportunity for people in the region to express their view.
Russia previously used a referendum as a pretext for annexation in Ukraine's Crimea in 2014, which the international community has not recognised.
Putin maintains Russia is carrying out a "special military operation" to demilitarise Ukraine, rid it of dangerous nationalists and defend Russia from transatlantic alliance NATO.
