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