This new development was confirmed both by Anatoly Kucherena, his legal representative in Russia, and by Wikileaks via their Twitter feed.
According to local reports, Snowden left the airport at about 15:30 local time after Kucherena handed him the asylum documents released by the Russian Federal Migration Service.
He apparently took a taxi - it's still unclear whether by himself or in the company of WikiLeaks' Sarah Harrison - and went in an unknown direction.
“He is the most wanted person on earth and his security will be a priority,” Kucherena explained. “He will deal with personal security issues and lodging himself. I will just consult him as his lawyer.”
While his location will likely remain secret for the time being, Snowden has expressed his intention to talk to the press in Russia after he has had at least one day to himself.
The papers that were issued to him are valid for a year, so he will have to reapply for asylum once that period nears the end.
"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. I thank the Russian Federation for granting me asylum in accordance with its laws and international obligations," Snowden stated after he received the asylum papers.
Russian president Putin has stated that they would not extradite Snowden to the US, but has asked him not to leak any more documents if he wanted to stay in Russia. It has not been specified whether this includes the still unpublished documents he previously shared with The Guardian's journalists.