Chasing 145 for victory on a turning track with variable bounce, India overcame a batting collapse to reach the target in the first session on the penultimate day at the Shere Bangla National Stadium.
Ravichandran Ashwin top-scored for India with an unbeaten 42, while off-spinner Mehidy Hasan (5-63) was the pick of the Bangladesh bowlers.
Having suffered a top-order meltdown on Saturday, India resumed on 45-4 needing 100 runs to sweep the series and boost their chances of making the final of the World Test Championship.
They slumped to 74-7 before Ashwin and Shreyas Iyer combined in an unbroken 71-run partnership to maintain India's unbeaten test record against Bangladesh.
Ashwin, dropped on one by Mominul Haque at short leg, hit Mehidy for successive boundaries to seal India's nervy win.
"There's always a belief that someone will put their hands up and do the job," stand-in India captain KL Rahul said.
"But it's also human to get nervous and have some nerves in the dressing room, and I'm not going to lie and say that there was no nerves in the dressing room and that everything was calm."
"We never thought it was going to be a cakewalk. We always knew it was going to be difficult and we had to really grind and work hard for those runs."
Bangladesh captain Shakib Al Hasan was pleased with his side's fighting spirit.
"We put on a good show, very good test cricket. That's what the crowd loves," Shakib said.
"The way we fought in the two test matches was impressive."
"I'm sure from here on we can only improve and, hopefully, next year it will be a different result for Bangladesh cricket, especially in test match(es)."
Ashwin, who also claimed a match haul of six wickets, was adjudged player-of-the-match.
India's Cheteshwar Pujara bagged the player-of-the-series award for being the leading scorer of the series.
Bangladesh had prevailed 2-1 in the preceding one-day international series between the neighbours.
