The incident that took place on Wednesday was the latest instance of a civilian vessel hitting an explosive in the Black Sea.
Ukraine has said increased Russian attacks on shipping and port infrastructure are to blame.
Russia's defence ministry did not immediately respond to a request for comment.
A shipping source speaking on condition of anonymity told Reuters the Greek-operated VYSSOS was the vessel that had been hit.
The vessel's Athens-based manager, which was listed in databases as Nava Shipping, could not be immediately located for comment.
"A Panama-flagged civilian vessel was blown up on an enemy sea mine in the Black Sea... The vessel lost its course and control, and a fire broke out on the upper deck," Ukraine's southern military command said on Telegram.
A captain and a sailor, an Egyptian citizen, were injured, and the latter was taken to hospital in the city of Izmail, the head of the Odesa regional prosecutor's office told a briefing.
President Volodymyr Zelenskiy says Ukraine's export corridor has had "significant results". (AP PHOTO)
He said the incident happened early on Wednesday at a river mouth.
The vessel stopped updating its position, according to the ship tracking and maritime analytics provider MarineTraffic.
It was heading to Izmail, the United Kingdom maritime security company Ambrey told Reuters.
The mine hit its stern, and detonation caused equipment and machinery failure.
There were 18 crew members onboard: 13 Egyptians, three Ukrainians and two Turkish citizens.
Ukraine says Russia has increased attacks on Ukraine's port infrastructure since mid-July, when it quit a United Nations-brokered deal that allowed safe passage of Ukrainian grain shipments via the Black Sea.
Ukraine since has established an alternative route, which hugs the western shores of the Black Sea.
Ukrainian President Volodymyr Zelenskiy said on Thursday that Ukraine's alternative export corridor had sent out 12 million tonnes of cargo so far.
He added in his nightly video address that the corridor had produced "particularly significant results for December, and this is felt at the level of our entire economy".
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