Relations between Russia and Georgia have been strained in recent months, since Russia stepped up support for Abkhazia and sent more peacekeeping troops into the region. Tbilisi has accused the Kremlin of trying to annex the territory.
Bagapsh said prior to his visit that Russia provides significant humanitarian aide to Abkhazia helping the republic to repair its infrastructure.
The president also said that Russian energy giant Gazprom is developing a program to provide natural gas supplies to Abkhazian consumers.
Moscow has repeatedly dismissed Tbilisi's claims that it is seeking to annex Abkhazia and has condemned Georgia's policy toward the breakaway region, which Russia says could lead to new bloodshed.
Abkhazia broke away from Georgia in the early 1990s following the collapse of the Soviet Union. Between 10,000 and 30,000 people were killed in the subsequent hostilities. The two sides signed a ceasefire in 1994 in Moscow.
Georgia under the Western-leaning President Mikheil Saakashvili is seeking NATO membership, which has also fueled tensions between Moscow and Tbilisi.