In Washington, the White House said it was extremely disappointed about Russia's decision.
Spokesman Jay Carney said the White House was reevaluating whether a planned fall summit with President Barack Obama and Russian President Vladimir Putin should still take place.
U.S. State Department spokeswoman Marie Harf explained Carney’s remarks to reporters in Washington:
"We do have a broad agenda that we talk to the Russian government about," Harf said. "There are issues where we work together, as you know: Afghanistan, Iran sanctions, and elsewhere. But there are times when we very strongly disagree."
"I think that today’s action, as my colleagues at the White House have said, is extremely disappointing," Harf continued. "And so, in light of the fact that they have taken such action, it behooves us to evaluate where the relationship is, whether the summit makes sense.”
Harf said U.S. officials were not informed in advance about Russia’s decision to grant Snowden asylum -- a move that allowed Snowden to obtain the documents needed to leave the airport transit area where he had been stranded since June 23, when the U.S. revoked his passport.
But she also said Washington and Moscow should continue to work together on issues where it is in the interests of both countries to do so:
"I think we and President Putin himself have been very clear that we don’t want this issue to broadly negatively affect our bilateral relationship because there are places where we work together, including in Afghanistan, with Iran sanctions, with reduction in our nuclear arms arsenals," she said.
"So we’ve both been very clear that this is an example of something we want to treat separately, that we don’t want it to adversely affect our whole relationship. That being said, this was a truly disappointing step."
Snowden's lawyer Anatoly Kucherena said Snowden left the transit zone of a Moscow airport on August 1 and officially entered Russia.
He would not specify Snowden’s whereabouts, saying he was now the most hunted man in the world and that it would be dangerous to talk about his location.
Snowden’s one-year asylum can be extended indefinitely. He also has the right to seek Russian citizenship.
Washington wants Snowden extradited from Russia so he can be put on trial for espionage after he leaked information to journalists about large-scale U.S. Internet- and phone-surveillance programs.
Amnesty International Russia Director Sergei Nikitin has welcomed Moscow's move to grant Snowden asylum, saying it was one of those "very rare" occasions when the rights watchdog was in agreement with Russia's authorities.
"I think in this concrete situation, as we hear accusations against Snowden from very high-profile people in the United States, as he has been already labeled a traitor, I think that in this situation he is obviously safer, at this stage, in the Russian Federation, as paradoxical as it sounds," Nikitin said.
Meanwhile, the founder of Russia's most popular social network, VKontakte, publicly offered a job to Snowden.
Pavel Durov, the 28-year-old co-founder of VKontakte, posted a message on his personal page on the site saying, "We invite Edward to [St.] Petersburg and will be happy if he decides to join the dream team of VKontakte programmers."