Assistant Sub-inspector M K Mazumdar (45) went into coma after the accident on June 24 and later underwent three surgeries but could not come out.
"He succumbed to his injuries this morning at a hospital here," a senior BSF officer said.
This is the second death of BSF official in the last 40 days in attacks along the Bangla border.
Mazumdar hailed from Nadi and was on duty in border post Kajuri of North 24 Parganas district of the West Bengal when his party detected a vehicle carrying migrants coming illegally into India in the night of June 24.
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On August 4 this year, another Border Security Force (BSF) trooper ASI Rajender Singh who was attacked by miscreants from across the border had succumbed to his injuries.
This year, a total 50 BSF personnel have sustained serious injuries in such incidents.
Majumdar is survived by his wife and a daughter.