Mr Snowden, who worked for the CIA, leaked documents to journalists showing the extent of the NSA's snooping programmes.
The US has labelled Russia's decision to grant asylum to fugitive intelligence leaker Edward Snowden as "extremely disappointing".
The White House is reconsidering a meeting scheduled for next month between President Barack Obama and Russian counterpart Vladimir Putin.
The US wants Mr Snowden extradited and tried for leaking secrets.
Mr Snowden, who left a Moscow airport transit zone for the first time since June, thanked Russia for its action.
'Stab in the back'
Mr Obama had been scheduled to meet Mr Putin on the sidelines of a G20 summit in early September in St Petersburg.
However, White House spokesman Jay Carney said: "We're extremely disappointed that the Russian government would take this step despite our very clear and lawful requests in public and in private to have Mr Snowden expelled to the United States to face the charges against him.
"We're evaluating the utility of a summit in light of this and other issues."
Democrat Senator Chuck Schumer, an ally of Mr Obama, said Russia's decision was a "stab in the back" and urged the president to recommend relocating the G20 summit away from Russia.
Former Republican presidential candidate John McCain said there should be "serious repercussions".
Russia has consistently played down the importance of the issue and insisted relations with the US should be unaffected.
The BBC's Daniel Sandford in Moscow says relations between the US and Russia were already bad and, after what appears to be a deliberate move by Mr Putin, they have now taken a definite turn for the worse.
Moscow and Washington are already at loggerheads on a number of issues, particularly the Syria conflict, where Russia supports the government and the US backs the rebels.
Mr Snowden, who worked for the CIA and later for a firm contracted to the National Security Agency, leaked documents to journalists showing the extent of the NSA's snooping programmes.
The stream of stories based on his leaks continued on Thursday with the UK's Guardian newspaper alleging the US government paid at least £100m ($150m) to the UK's GCHQ spy agency to secure access to Britain's spy programmes.
The fugitive left Sheremetyevo Airport at about 14:00 local time (10:00 GMT), according to his lawyer Anatoly Kucherena.
Mr Kucherena said his client "assured me that he is not planning to publish any documents that blacken the American government".
Mr Putin had earlier said this was a condition for granting any asylum.
Mr Kucherena said Mr Snowden had been given shelter in a private home of American expatriates.
Russia's Federal Migration Service later confirmed that Mr Snowden had been given asylum for one year.
Mr Snowden's presumed final destination is Latin America, but experts were unsure whether he would be able to travel abroad with the Russian travel documents.
Mr Snowden released a statement after leaving the airport thanking Russia, and accusing the US of showing no respect for the law.
Information leaked by Mr Snowden first surfaced in the Guardian and Washington Post newspapers in early June.
It showed that the NSA was collecting the telephone records of millions of Americans.
The systems analyst also disclosed that the NSA had tapped directly into the servers of major internet firms to track online communication in a surveillance programme known as Prism.
Allegations later emerged that the US had spied on its EU allies, and ran huge data-collection operations from Brazil that snooped on Latin American businesses and governments.