Volunteers help people arriving from Ukraine, at Berlin's central station in Berlin, Germany. File/Reuters
UNHCR, the UN's refugee agency, recorded 2,011,312 refugees on its dedicated website, 276,244 more than the previous count on Monday.
UNHCR chief Filippo Grandi called it a "shocking milestone".
"Behind the monolithic statistics are two million stories of separation, anguish, and loss," he said.
People wait inside the main hall of the railway station in Przemysl, Poland. File/Reuters
Families have been "senselessly ripped apart", plunged into "despair and unimaginable suffering" by the "brutal war", he said.
Authorities and the UN expect the flow to intensify as the Russian army advances deeper into Ukraine, particularly as it approaches the capital, Kyiv.
Before Russia invaded, more than 37 million people lived in Ukrainian territory under the control of the central government.
Besides those who have left, an unknown number have been displaced from their homes within the country.
The International Organization for Migration said that 103,000 third-country nationals were among those who have fled.
"There are countless tens of thousands of others who remain in the country stranded," IOM spokesman Paul Dillon said, citing a mixture of overseas students and workers.
He announced a partnership with Airbnb to connect refugees to free or heavily discounted short-term housing in neighbouring countries, with more than 26,000 hosts having signed up so far.
"The real toll is likely to be much higher," Liz Throssell, a spokesperson for the UN human rights office (OHCHR), told a briefing, adding that 253 of the casualties were in the Donetsk and Lugansk regions in eastern Ukraine.
"To the world: what is the point of saying 'never again' for 80 years, if the world stays silent when a bomb drops on the same site of Babi Yar," Ukrainian leader Volodymyr Zelensky asked in a tweet.
The United Nations said more than 660,000 refugees had fled to neighbouring countries since Russian President Vladimir Putin launched what he has called "a special military operation" last Thursday.
Asad Kaleem, an executive producer at ARY, said that the action means that 4,000 employees at the TV are now without work. He pleaded with the government to reverse its decision and bring the hugely popular ARY back on air.
The cabinet shake-up, which was approved by parliament in an emergency session, affected 13 portfolios, including health, education, culture, local development and irrigation ministries.
Other infrastructure was also damaged when gusts battered the Medusa Festival, a huge electronic music festival held over six days in the east coast town of Cullera.