A BSF sector commander ranked official said Prashanta Roy drowned when he was swimming to get hold of a smuggler, who jumped into a local river which separates the two countries.
As Roy raised an alarm, local people and jawans dived into the river and brought him up.
However, he was declared dead at Balurghat Hospital.
The rifle of the jawan, from which he had fired one shot during the chase, was also recovered from the water, the official said.