Besides scoring 103, Harmanpreet also took two wickets, as India restricted Bangladesh to 210 for nine after putting up 256 for six on the board. The Indians elected to bat after winning the toss.
The hosts were going steady at 75 for two in the 20th over when Harmanpreet arrived at the crease.
She shared 109 runs off 126 balls for the third wicket with opener Poonam Raut, laying the foundation for a competitive total.
The ton-up batswoman was dismissed in the 48th over by Salma Khatun, soon after bringing up her second ODI hundred, which helped India set Bagladesh a competitive target.
Harmanpreet's 103 came off 100 balls and included 11 fours and two sixes. She also became the sixth Indian batswoman to score more than 1000 runs in ODIs.
Needing to score at around five runs an over, Bangladesh could not get the momentum going their way.
Ahmed's dismissal in the 35th over, with 114 needed off 15 overs, triggered off a collapse and Bangladesh lost the next seven wickets for 56.
Ahmed top-scored for Bangladesh with a 94-ball 75.
Ekta Bisht was the leading wicket-taker for India, picking up her second consecutive three-wicket haul to finish with figures of three for 34 in 10 overs.
Harmanpreet picked up two wickets for 30 runs off her six overs.
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