Ravichandran Ashwin was the only bright spot on another wet and rain-curtailed day as the off-spinner completed his 10th five-wicket haul in tests.
Bangladesh were bundled out for 256 in their first innings and with 30 overs left in the final session, Tamim Iqbal (16) and Imrul Kayes (seven) batted out 15 overs to reach 23-0 before the teams shook hands for a draw.
A steady drizzle washed out the morning session as Bangladesh, resuming on 111-3, continued to bat positively with little time left in the test match.
Ashwin, who had picked up two wickets on Saturday, dismissed Shakib Al Hasan (nine) in the second over of the day when the left-handed batsman got a thick top edge to wicketkeeper Wriddhiman Saha while attempting a cut shot.
Opener Imrul Kayes (72) was beaten by the flight and bounce from experienced Harbhajan Singh who got the left-hander stumped for his second wicket in his first match in over two years for India. The 34-year-old finished with 3-64 with the wicket of Mohammad Shahid towards the end.
Kayes and Soumya Sarkar (37), who chopped on Varun Aaron to give the only wicket to a paceman in the innings, added 51 runs for the fifth wicket.
Shuvagata Hom (nine) became Ashwin's fourth victim and fell in the final delivery before the tea break, getting caught at short leg.
Debutant wicketkeeper Litton Das played a quickfire knock of 44 from 45 balls before giving Ashwin his fifth wicket for his first five-wicket haul in an innings outside India.
The teams will now play a three-match one-day international series, with the first match in Dhaka on Thursday.
(Writing by Sudipto Ganguly in Mumbai; editing by Julian Linden and Pritha Sarkar)
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