The port, which had recently been occupied by Russian forces with several Russian warships in dock, was rocked by a series of heavy explosions soon after dawn on Thursday.
Social media videos showed fires raging at the dockside, with a series of secondary explosions reverberating across the city.
The Ukrainian armed forces on Friday named the ship as the "Saratov." In earlier reporting, the ship was named as the "Orsk."
In a statement, the armed forces said: "In the Azov operational zone, according to updated information, a large landing ship "Saratov" was destroyed during the attack on the occupied Berdiansk port. Large landing ships "Caesar Kunikov" and "Novocherkassk" were [also] damaged. Other losses of the enemy are being clarified."
Several Russian ships had been unloading military equipment at Berdyansk in recent days, according to reports from the port by Russian media outlets.
The United States said that Ukraine likely did conduct a successful attack against Russian ships in Berdiansk, according to a defense official, though it is unclear what type of weapon or weapons were used in the attack. It echoes a similar statement from the British Ministry of Defence, which said that Ukrainian forces have attacked "high value targets" in Russian-occupied areas of Ukraine, including a landing ship and ammunition depots at Berdyansk.
Analysis of videos uploaded on Thursday showed that one Russian naval vessel left the port soon after the explosions.
The Russian Ministry of Defense has made no official comment about the explosion.
Berdyansk sits on the Azov Sea and is roughly 45 miles (70 kilometers) southwest of Mariupol. The city has a small naval base and a population of about 100,000.
Russian military troops first occupied Berdyansk government buildings on February 27, three days after Russia's invasion began.
Mariupol still eludes Russian control despite being surrounded and mercilessly pummeled, block by block, by Russian firepower.
Its defenders rejected an ultimatum to surrender by Monday morning, thwarting a Russian effort to finalize a land bridge linking Crimea with the separatist republics of the eastern Donbas region.
Russia has fired on Mariupol from the Sea of Azov, according to a senior US defense official, using a group of approximately seven ships to launch attacks on the critical coastal city.
Further west, Ukrainians have been fighting to take back the city of Kherson, as well as pushing Russian forces from the northeast of Mykolaiv, forcing them to reposition south of the city, a senior US defense official said Tuesday.
The official cautioned that the US cannot say whether these moves are part of a "larger operational plan" by the Ukrainians, but called the Ukrainian defense "nimble" and "agile."