Alexander Lukashevich of the Foreign Ministry said Russia is "ready for close cooperation" with Annan in efforts to find a solution to the Syrian crisis, RIA Novosti reported.
Russia previously vetoed several efforts to pressure Syrian President Bashar Assad's regime to end the nearly year-long violence against demonstrators.
"We expect that the activity of this authoritative politician [Annan] will contribute to solving the pressing political and humanitarian issues in Syria on the basis of work with all parties in the interest of peace through inclusive political dialogue between the Syrian government and the opposition," Lukashevich said in a statement.
In a joint statement, U.N. Secretary-General Ban Ki-moon and Arab League Secretary-General Nabil El-Araby said Annan will serve as their representative on the crisis and will have a deputy from the Arab world to be announced later, U.N. News said on its Web site.
The announcement said the crisis in Syria has claimed the lives of thousands of people in the government's crackdown on the pro-democracy uprising.
"The special envoy will provide good offices aimed at bringing an end to all violence and human rights violations, and promoting a peaceful solution to the Syrian crisis," the statement said.
Annan will "consult broadly and engage" with relevant parties inside and outside of Syria to find "a peaceful Syrian-led and inclusive political solution that meets the democratic aspirations of its people," the Ban-El-Araby statement said.
Annan was U.N. secretary-general from 1997 through 2006.
The new appointment was announced after a U.N.-appointed panel issued a report saying the human rights situation inside Syria deteriorated "significantly" since November, with ongoing widespread, systematic and gross violations by security forces against civilians.
The panel reported in November Syria's military and security forces had committed crimes against humanity with their crackdown on protesters. The report documented patterns of summary execution, arbitrary arrest, enforced disappearance, torture, including sexual violence, and violations of children's rights, U.N. News said.
In its latest report, the three-member panel said the Syrian crisis had become "increasingly violent and militarized" with the rise of an armed opposition, with no let up in the crackdown on peaceful protesters.
Ban also asked Valerie Amos, undersecretary-general for humanitarian affairs, to visit Syria to assess the humanitarian situation, U.N. News said.