Grainy images on Russian television showed Snowden's new document, which is similar to a Russian passport, and revealed that he had been granted asylum for a year from July 31.
"He is the most wanted man on planet Earth. What do you think he is going to do? He has to think about his personal security. I cannot tell you where he is going," his lawyer, Anatoly Kucherena, told Reuters.
"I put him in a taxi 15 to 20 minutes ago and gave him his certificate on getting refugee status in the Russian Federation," he said. "He can live wherever he wants in Russia. It's his personal choice."
He said Snowden, who had his U.S. passport revoked by Washington after he fled to Moscow from Hong Kong on June 23, was not going to stay at an embassy in Moscow, although three Latin American countries have offered to shelter him.
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Snowden, 30, was accompanied by Sarah Harrison, a representative of the anti-secrecy group WikiLeaks, which confirmed he had left the airport.
"We would like to thank the Russian people and all those others who have helped to protect Mr. Snowden. We have won the battle - now the war," WikiLeaks said on Twitter.
Nicaragua, Bolivia and Venezuela have offered Snowden refuge, but there are no direct commercial flights to Latin America from Moscow and he was concerned the United States would intercept his flight to prevent him reaching a new destination.
He was forced to bide his time in the transit area between the runway and passport control, which Russia considers neutral territory. Kucherena had given Snowden Russian books to help pass the time and says he has started learning Russian.
Strains in ties
The White House has signalled that President Barack Obama might consider boycotting a planned summit with Russian President Vladimir Putin in Moscow in early September over the Snowden case.
But a senior Kremlin official played down concerns.
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"Our president has ... expressed hope many times that this will not affect the character of our relations," Yuri Ushakov, Putin's top foreign policy adviser, told reporters shortly after news of Snowden's departure from the airport.
He said there was no sign that U.S. President Barack Obama would cancel the planned visit in September.
It is not clear what Snowden plans to do in Russia, although he has said he would like to travel around the country.
More than half of Russians have a positive opinion of Snowden and 43 percent wanted him to be granted asylum, a poll released by independent research group Levada said this week.
Snowden's arrival at Sheremetyevo put Putin in an awkward position. He has said he does not want the case to undermine relations with Washington but would have risked looking weak if he had handed him over to the U.S. authorities.
Both Russia and the United States have signaled they want to improve ties, strained by issues ranging from the Syrian conflict to Putin's treatment of opponents and Western-funded non-governmental organisations since he started a third term in 2012.
Putin has said Snowden must stop anti-U.S. activities, but it was not clear whether the American had agreed to do so.
Snowden has said previously that he does not regard his activities as hostile to the United States, although Kucherena has said his client has agreed to halt such actions.
There has already been diplomatic fallout from Snowden's leaks, which included information that the U.S. National Security Agency bugged European Union offices and gained access to EU internal computer networks, even though the EU is an ally.
China, Brazil and France have also voiced concern over the spying program.
U.S. relations with Latin American states have been clouded by the refusal of four U.S. allies in Europe to let a plane carrying Bolivia's president home from Moscow use their airspace because of suspicion that Snowden might be on the plane.