The bombs went off with an interval of five minutes at 8:10 a.m. Moscow time (4:10 GMT) on Sunday at the Gagry marketplace and near the Continent supermarket, the police said.
Six local residents were hit by the first bomb on the city marketplace while the second bomb caused no damage. No Russian citizens having their vacation in Abkhazia have been hurt in the incident, the police said.
An investigation is under way, the police said.
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 Georgian-Abkhazian hostilities.
The pro-Western Georgian government of Mikheil Saakashvili has said it is determined to bring the breakaway region back under its control.