His whereabouts will be kept a secret for security reasons, his lawyer, Anatoly Kucherena, said, making it even harder to keep track of the former systems analyst with the United States National Security Agency.
The 30-year-old had been largely hiding out at Sheremetyevo airport, since his arrival from Hong Kong on 23 June.
The US said it was “extremely disappointed” in Russia’s decision to grant him asylum.
The White House warned it was not a positive development for US-Russia relations and said it undermined Russia’s record of law enforcement co-operation with the Americans.
It said it was now re-evaluating whether President Barack Obama should attend a forthcoming summit with Russian counterpart Vladimir Putin.
“We are extremely disappointed that the Russian government would take this step despite our very clear and lawful requests in public and private that Mr Snowden be expelled and returned to the United States,” said White House spokesman Jay Carney.
The US had demanded that Russia send Snowden home to face prosecution for espionage over leaks that revealed internet surveillance practices, but Mr Putin dismissed the request.
In a statement released through Wikileaks, Mr Snowden thanked Russia and criticised the Obama administration.
“Over the past eight weeks, we have seen the Obama administration show no respect for international or domestic law, but in the end the law is winning,” he said. “I thank the Russian federation for granting me asylum in accordance with its laws and international obligations.”
Mr Putin has said his asylum was contingent on him not hurting US interests, but the Kremlin could have interpreted that to exclude documents he had already leaked to newspapers that continue to trickle out.
In his application for asylum, Mr Snowden said he feared he could face torture or capital punishment if he was returned to the US, though the White House had promised Russia that would not be the case.
The US has revoked his passport, and the logistics of him reaching other countries that had offered him asylum, including Venezuela, Nicaragua and Bolivia, are complicated.
“He now is one of the most sought-after men in the world,” Mr Kucherena said. “The issue of security is very important for him.”
New revelations about US intelligence-gathering, based on information from Mr Snowden, was published earlier this week.
However, Mr Kucherena said the material had been provided before Mr Snowden promised to stop leaking as a requirement of getting asylum.
The one-year asylum can be extended indefinitely, and Mr Snowden also has the right to seek Russian citizenship.
According to the rules set by the Kremlin, a person with temporary asylum would lose it if he or she travelled abroad.
Mr Kucherena said it would be up to Mr Snowden to decide whether to travel to any foreign destination, but he added that “he now has no such plans”.
Mr Snowden’s father has said he wants to visit his son, and Mr Kucherena is arranging the trip.
Wikileaks, which has adopted Mr Snowden’s cause, said legal adviser Sarah Harrison, who had been in the transit zone for 39 days, was still with him. It said only that they left the airport in a taxi and were heading to a “secure, confidential place”.
The group also praised Russia for providing him with shelter, adding: “We have won the battle – now the war.”