There were protests against the elections in Tbilisi, the Georgian capital.
A few hundred protesters also demonstrated outside Abkhazia, camping overnight around a bonfire, local reporters at the scene said.
"Despite the fact that Georgia tried everything to frighten the local population in Gali, people are going to the polls and participating"
Earlier Abkhaz officials threatened that their forces would fire on the demonstrators if they crossed a river into Abkhaz territory.
Bagapsh travelled to Gali, an area of Abkhazia that borders Georgia and has a largely Georgian population, to oversee voting.
He was reported by the Interfax news agency as saying: "Despite the fact that Georgia tried everything to frighten the local population in Gali, people are going to the polls and participating."
But Paata Shamugia, a protest leader and an official appointed by Tbilisi for the Gali district, disputed that the voting was going ahead successfully.
He was reported by AFP as saying: "No-one is voting, neither Abkhaz or Georgians. Bagapsh had to go to Gali to raise the level of activity."
Giga Bokeria, a Georgian parliamentary deputy and a close ally of Mikhail Saakashvili, the Georgian president, said that the election was "not legitimate".
He said: "We are worried about the fate of Abkhazia and especially those who live in Gali."
Abkhazia is one of two separatist territories in Georgia that have de facto backing from Russia, although Moscow does officially recognise Georgia's territorial integrity.
Saakashvili says the territories are being artificially sustained by Moscow in an attempt to maintain Russian influence in the ex-Soviet region.
Georgia has promised to regain control of Abkhazia, which broke away from the ex-Soviet republic after a 1992-1993 war in which several thousand people died and 250,000 people were forced from their homes.
Georgian troops fled the province at the close of the war and Abkhazia later declared independence.
The territory has not been recognised internationally, with the EU saying Georgia should not be broken up.