The anti-secrecy group WikiLeaks said Snowden was heading for a "democratic nation" which it did not name, although a source at the Russian airline Aeroflot said he would fly on within 24 hours to Cuba and then planned to go to Venezuela.
Moscow airport officials said the flight from Hong Kong had landed but could not immediately confirm Snowden was on board. However, a source at Aeroflot said he had booked a seat on the service.
Snowden, who worked for the National Security Agency, had been hiding in Hong Kong since leaking details about U.S. surveillance activities to news media.
A spokesman for the government of Hong Kong, a former British colony which returned to China in 1997, said it had let Snowden depart because a U.S. request to have him arrested did not comply with the law.
The United States wanted him to be extradited to face trial and is likely to be furious about his departure. In Washington, a Justice Department official said it would seek cooperation with countries Snowden may try to goto.
"It's a shocker," said Simon Young, a law professor with Hong Kong University. "I thought he was going to stay and fight it out. The U.S. government will be irate."
A source at Aeroflot said Snowden would fly from Moscow to Cuba on Monday and then planned to go on to Venezuela. The South China Morning Post earlier said his final destination might be Ecuador or Iceland.
The WikiLeaks anti-secrecy website said it helped Snowden find "political asylum in a democratic country".
It added in an update on Twitter that he was accompanied by diplomats and legal advisers and was travelling via a safe route for the purposes of seeking asylum.
"The WikiLeaks legal team and I are interested in preserving Mr Snowden's rights and protecting him as a person," former Spanish judge disclosures in the public interest - is an assault against the people."
Assange has taken sanctuary in the Ecuadorean embassy in London and said last week he would not leave even if Sweden stopped pursuing sexual assault claims against him because he feared arrest on the orders of the United States.
U.S. authorities have charged Snowden with theft of U.S. government property, unauthorized communication of national defense information and wilful communication of classified communications intelligence to an unauthorized person, with the latter two charges falling under the U.S. Espionage Act.
The United States had asked Hong Kong, a special administrative region (SAR) of China, to send Snowden home.
"The U.S. government earlier on made a request to the HKSAR government for the issue of a provisional warrant of arrest against Mr Snowden," the Hong Kong government said in a statement.
"Since the documents provided by the U.S. government did not fully comply with the legal requirements under Hong Kong law, the HKSAR government has requested the U.S. government to provide additional information ... As the HKSAR government has yet to have sufficient information to process the request for provisional warrant of arrest, there is no legal basis to restrict Mr Snowden from leaving Hong Kong."