EU report accuses both sides of violating international law in last year’s conflict.
Russian election observers said voting was “fair and transparent” while Georgia has called the poll “illegitimate and amoral”.
Five candidates were in the running for the leaderhip of Abkhazia, which was formally recognised as an independent state by Moscow after a brief war with Georgia in August last year.
‘Flashpoint for unrest’
Abkhazia, which broke away from Georgia in 1993 after a 13-month war, is considered to be a flashpoint for further unrest between Russia and Georgia in the South Caucasus.
The vote comes five years after a presidential election in late 2004 which saw Khadzimba challenge a result that was in Bagapsh’s favour.
Bagapsh has won support from some Abkhaz after winning Russia’s recognition of the territory as an independent state during his rule.
Khadzimba had tried to appeal to those who say that Bagapsh has given too much influence to Russia in Abkhaz affairs.
Abkhazia and South Ossetia, another breakaway region in Georgia, rejected Tbilisi’s rule soon after the break-up of the Soviet Union in 1991.
In August 2008, Georgia launched an assault on South Ossetia that brought a counter-strike by Russian forces. Abkhazia took back its last enclave held by Georgia during the conflict.
About 3,600 Russian soldiers are deployed in Abkhazia, where two Russia military bases are under construction.
Nicaragua and Venezuela are the only countries apart from Russia that recognise Abkhazia as an independent state.