Neighbourly relations have been strained by the student-led uprising that toppled Bangladesh's leader Sheikh Hasina, who fled to India on August 5.
Since Hasina's ousting, Indian security forces have detained or sent back dozens of Bangladeshis accused of trying to sneak across the border.
Indian border officers say the teenage girl was found with a bullet wound during a routine patrol in Tripura state on Sunday, but gave no further details.
Bangladeshi police have accused Indian officers of killing the girl, who was shot in the chest, according to Binoy Bhushon Roy, a police officer in the Moulvibazar border district.
Himanta Biswa Sarma, chief minister of India's Assam state, said police managed on Thursday to "push back" five more Bangladeshis who had tried to cross.
"We are committed to an infiltration-free Assam," Sarma said in a statement, praising police for a "constant vigil along the Indo-Bangladesh border".
Bangladesh is almost entirely encircled by India, and their shared border stretches more than 4,000 kilometres (2,485 miles).
Last month, security forces from India and Bangladesh had a tense confrontation over the construction of a fence supposedly meant to keep cattle from straying over the border.