Russia's ITAR-TASS news agency says one person was wounded during the melee.
Prime Minister Nodar Khashba had left the building earlier in the day. His whereabouts and that of President Vladislav Ardzinba are not known.
The demonstrators were protesting against Ardzinba's decision to order a new vote early next month in place of the disputed 3 October presidential election.
Election officials had declared opposition contender Sergei Bagapsh the winner of the polls. But the outgoing administration had ruled against the decision, claiming the vote was fraudulent.
[For the latest news on the tensions in Abkhazia, see RFE/RL's webpage on Abkhazia and Georgia.]