Skipper Suresh Raina was the highest scorer for India as he scored 27 off 23 balls.
Robin Uthappa scored 14 off 24 balls before being dismissed by Taskin Ahmed.
Taskin Ahmed claimed a five-wicket haul as Bangladesh bowled India out for 105 runs in a rain-marred 41-over match.
Ambati Rayudu played seven deliveries and scored just 1 run.
Mashrafe Mortaza claimed two wickets in his spell.
Mohit Sharma picked up four wickets as India registered a 47-run over win over Bangladesh in the 2nd ODI.
Stuart Binny's 6 for 4 was best figures for an Indian in the 50-over format as Bangladesh equalled their lowest ODI total.