They have now won 18 home Test series in a row, beating the previous record they held, which was brought up by a record-breaking performance with the bat on day four.
Jasprit Bumrah, Ravichandran Ashwin and Ravindra Jadeja each claimed three wickets as Bangladesh were skittled out for 146 before lunch on Tuesday, leaving India needing 95 to win.
In a contest that was postponed for two and a half days due to rain, India easily defeated their goal with to opener Yashasvi Jaiswal’s 51, his second half-century of the game.
They now have an eight-point advantage over Australia at the top of the World Test Championship.
India were the first side to reach 50, 100, 150, 200, and 250 in Test matches, setting the stage for victory on a record-breaking fourth day.
India is playing its first Test series since Gautam Gambhir, a former opener, was named head coach.
Now, a three-match Twenty20 series between the two teams will begin on Sunday in Ghana.