One Indian guard was also wounded, Indian officials said.
The fisherman had been held because he was fishing in the Padma River at the border, where Bangladesh has banned fishing, it said.
On Friday, the situation at the border was "normal", Shariful Islam a spokesman for the Bangladeshi force said.
"Our troops have been put on high alert to avoid any untoward incident," he said.
Bangladesh officials said they would hand over the fisherman during a meeting later. Both sides have agreed to investigate the matter, Bangladesh said.
India and Bangladesh share close ties and struck a series of deals this month when Bangladeshi Prime Minister Sheikh Hasina visited New Delhi.
They share a border stretching more than 4,000 km (2,400 miles), where clashes have intermittently erupted over what India terms illegal immigrants from Bangladesh.
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