This was also the first loss of the women-in-blue in the tournament since 2012.
Fargana Hoque and Rumana Ahmed's unbeaten 52 and 42-run knocks helped Bangladesh chase a below par target of 142 with seven balls to spare.
Opting to bat after winning the toss, India started the proceedings on a poor note, losing openers Mithali Raj (15) and Smriti Mandhana (2) with the scorecard reading 26.
After their departure, Pooja Vastrakar chipped in with 20 runs before departing off a run-out in the ninth over.
Skipper Harmanpreet Kaur and Deepti Sharma then contributed with 42 and 32 runs respectively while Mona Meshram remained unbeaten on 14 to help India reach 141/7.
Bangladesh's leggie Rumana Ahmed bagged three wickets conceding 21 runs.
In reply, Bangladesh also lost couple of wickets for 45 runs.
But Fargana and Rumana denied India any further inroads, forging a match-wining 93-run third wicket partnership to help their side register their second win in the tournament.
Bangladesh had earlier beaten Pakistan in the tourney.
For India, Rajeshwari Gayakwad, Poonam Yadav and Vastrakar bagged one wicket each.
Despite the loss, India were static atop the points table with two wins from three games while Bangladesh is placed fourth.
The Harmanpreet-led side will next face Sri Lanka on Thursday.
In other league matches, Pakistan defeated Sri Lanka by 23 runs while Thailand thrashed Malaysia by nine wickets.
Fargana HoqueFargana Hoque and Rumana AhmedIndian women's teammithali rajRumana Ahmedsmriti mandhanawomens asia cup
First Published: June 6, 2018, 4:19 PM IST