MP Vakhtang Golandzia died of wounds sustained in the shooting at the parliament building, the press office of the acting president Badra Gunba said in a post on Telegram.
Another MP was wounded in the arm.
Abkhazian state news agency Apsnypress said the interior ministry had identified another MP, Adgur Kharazia, as the suspect in the shooting and that he had fled the scene.
Its report did not give any indication as to a motive.
A lush subtropical territory on the Black Sea coast, Abkhazia broke from Georgia's control in a war after the collapse of the Soviet Union in the early 1990s, during which hundreds of thousands of ethnic Georgians fled.
Russia has long supported Abkhazia and another breakaway Georgian region, South Ossetia, and recognised them as independent after winning a five-day war against Georgia in 2008.
Abkhazia was plunged into crisis last month when protesters stormed the parliament in opposition to an investment agreement with Russia, forcing the region's president, Aslan Bzhania, to resign.
His vice president, Gunba, became the interim leader.
Deputies voted against ratifying the agreement with Russia earlier this month.
Opposition leaders in Abkhazia had opposed the agreement over fears it would clear the way for wealthy Russians to buy up property, pricing out locals.
The Kremlin, which heavily finances Abkhazia's budget, wants Russian investors to be able to acquire property rights and to have the right to develop the region.
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