Mustafizur dismissed Rohit in the final over of the powerplay.
Dhawan was dismissed soon after.
India's in-form batsman Kohli struggled for timing on this surface.
Openers began in sedate fashion.
In the end, India were restricted to 146.
His stumping gave India a sniff.
Tamim made use of the dropped chance to hit a flurry of boundaries.
India struck early with Mohammad Mithun's wicket.
Hardik Pandya conceded two boundaries and Bangladesh were clear favourites to win the game with two needed off the last three balls.
He then dismissed both Mushfiqur Rahim and Mahmudullah to bring the equation down to two runs off the final ball.
R Ashwin got rid of Shakib al Hasan at a crucial time with Bangladesh still needing 52 off 47. However, Soumya Sarkar and Mahmudullah kept them in the game and brought the equation down to 11 off the last over.
After a long discussion, Pandya bowled a wide delivery that Shuvagata Hom failed to connect. Though the Bangladesh batsmen went for the bye, Dhoni out ran Mustafizur to run him out to seal a thrilling one-run win to keep their semifinal hopes alive.