The defending champions will take on arch-rivals Pakistan in the title clash on Sunday.
Needing 265 for victory, India chased down the target with 59 balls to spare at the Edgbaston.
Earlier, sent into bat by India skipper Virat Kohli, Bangladesh scored 264 for seven in the allotted 50 overs.
Brief scores:
Bangladesh: 264/7 in 50 overs (Tamim 70, Mushfiqur 61; Bumrah 2/40).
India: 266/1 in 40.1 overs (Rohit Sharma 123 not out, Virat Kohli 96 not out).
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