It was their best-ever performance in the tournament since making their Cup debut in England in 1999. They failed to go beyond the first round in England and then again in the next edition in 2003 in South Africa.
Bangladesh's second win in the four-team Group B also knocked former champions India out of the tournament. Sri Lanka have already advanced to the Super Eights with an all-win record.
India had already finished their group campaign with one win and two defeats and their slim hopes rested on an unlikely Bermudian win. The Cup debutants lost of their three matches.
Bangladesh had to face many anxious moments chasing a rain-revised target of 96 in the 21-overs-a-side game before entering the next round.
They survived a three-wicket burst from new-ball bowler Saleem Mukuddem before man-of-the-match Mohammad Ashraful (29 not out) and teenager Saqibul Hasan (26 not out) completed the job in semi-darkness at the Queen's Park Oval.
"We were a little bit nervous chasing the target as it was a very important match for us. Then, Ashraful and Saiqbul batted very well," said Bangladesh skipper Habibul Bashar.
"I am delighted as the boys showed a positive approach. I hope we continue to perform in the next round. We have been playing good cricket in the tournament. The win has given us confidence to do well in the Super Eights."
Mukuddem grabbed three wickets in his lively opening spell to send Bangladesh reeling at 37-3 in the pressure game, but he lacked support from the other end.
Saqibul sparked celebrations in the Bangladeshi camp when he lofted left-arm spinner Dwayne Leverock over mid-on for the winning four. He put on 59 for the unfinished fourth-wicket stand with Ashraful.
Left-arm spinner Abdur Razzak earlier bagged three wickets to raise Bangladesh's hopes of posting a crucial win. He finished with 3-20 to help his team restrict Bermuda to a modest total.
Fast bowler Mashrafe Mortaza and left-arm spinner Saqibul each took two wickets to provide valuable support.
Dean Minors top-scored for Bermuda with 23. Opener Oliver Pitcher made 22 and Lionel Cann smashed 16 off just six balls with one six and two fours. Pitcher was later stretchered off the field while attempting to take a catch.
Bashar won a good toss in overcast conditions and had no hesitation in putting Bermuda in to bat. The move clicked as Mortaza grabbed two early wickets to put the opposition in a spot.
Mortaza struck in his opening over when he had Stephen Outerbridge caught by Mohammad Rafique at fine-leg before accounting for Delyone Borden, who was held by Bashar at mid-off.
Bermuda suffered a big setback when they lost David Hemp (nought), the only professional in their ranks. Hemp, who cracked a half-century against India, was caught at Aftab Ahmed at point off left-arm seamer Syed Rasel.
The Cup debutants were reduced to 35-4 in the 11th over when Razzak trapped skipper Irvine Romaine leg-before for 11.