A spokesman for the interior ministry told Ukrainian television that one of the injured men had died in hospital. The other coastguards were still being treated but he declined to disclose the extent of their injuries.
"The search continues for one more person. It has been established that he is the cutter's commander," the State Border Guard Service said in a statement.
The eastern outskirts of the industrial Sea of Azov port, most of it controlled by Kiev's forces, has been under attack from pro-Russian militias for weeks.