The explosion on Friday was the second major incident in a populated area in just a week as the spillover threat increases from the war in Ukraine, which has been fighting against a Russian invasion since 2022.
Romanian officials did not specify whether the drone was of Ukrainian or Russian origin.
News website G4media cited Constanta county prefect Adrian Teodor Picoiu saying Ukraine had notified Romania that the drone was part of a group of five, with one exploding in Ukraine.
Authorities in Ukraine did not immediately comment, while Russia's embassy in Romania said the drones were Ukrainian naval unmanned aerial vehicles.
The explosion comes a week after authorities said a Russian drone crashed into an apartment building in the southeastern Romanian city of Galati, near the border with Ukraine, injuring two people - the first time in the war that a drone struck a densely populated area in a NATO state.
Romania's defence ministry said earlier the object was not part of the Romanian army's equipment nor involved in recent Black Sea exercises.
The port, Romania's largest, was evacuated, residents were warned to avoid the Black Sea coast, while ships and two helicopters were surveying the area, deputy Interior Minister Raed Arafat told a briefing.
European Union member Romania shares a 650km land border with Ukraine and has seen Russian drones repeatedly breach its airspace since Moscow attacked Kyiv four years ago, as well as mines floating in the Black Sea across key trade and energy routes.
Romania's navy on Wednesday detonated a Russian YaRM-type anti-landing mine that had drifted to its Black Sea shore.
"This is the second significant security incident this week on the Romanian seaside," Romanian President Nicusor Dan said on Facebook just after news of the detonation.
European Commission President Ursula von der Leyen said on X that Russia's war in Ukraine was increasingly becoming a direct threat to countries on Europe's eastern border.
The drone detonated some 500 metres away from an oil terminal.
The Black Sea is crucial for shipments of grain, oil and oil products and is shared by Bulgaria, Romania, Georgia and Turkey, as well as Ukraine and Russia.
