Russian Foreign Minister Sergei Lavrov said on Monday that Moscow would actively support the Palestinian president and develop Russian-Palestinian relations.
Abbas said the Palestinians had "high hopes" of diplomatic support from Moscow as efforts to restart top-level peace negotiations between the Palestinians and the Israelis continue.
Abbas was scheduled to meet President Vladimir Putin and other top Russian officials later on Monday.
His trip to Moscow "testifies to the important role" played by Moscow in efforts to bring about a peaceful resolution of the Israeli-Palestinian conflict as Russia, along with the European Union, the United Nations and the United States, is a member of the Middle East peace quartet", he said.
The centrist Russian newspaper Izvestia said Abbas was in Moscow to secure Russia's support, but added that his success in doing so was uncertain.
"The main goal of the Palestinian visitor is to obtain Russia's support in anticipation of difficult negotiations with the Israelis," the newspaper said.
"Is Moscow ready at this time to support the representative of the late Arafat?"
Russian daily Izvestia
"But the question remains open: Is Moscow ready at this time to support the representative of the late Arafat - not just with words but with concrete deeds?"
The paper said Abbas was in "urgent need" of strong outside diplomatic support and said his effort to win backing from Moscow was part of a larger campaign to fortify his international standing before facing Israel and its main ally, the United States, in a possible new round of peace talks.