The incident took place at Garden Reach area near the port around 1:30 am, they said.
The inspector, Gobindo Seal, was allegedly shot dead by the constable Neelkanta Behra, after Seal rebuked him for not following instructions during a roll-call parade.
Later, Behra went to Seal's quarter in the wee hours and shot him, police said.
"We have arrested the constable," DC (PORT) V Solomom said.