Razzak finished with 3-20, while fast bowler Mashrafe Mortaza and left-arm spinner Saqibul Hasan each took two wickets in a match reduced to 21-overs-a-side due to rain.
Bangladesh, who had beaten India in their first match, needed a win to make it to the second round for the first time since making their Cup debut in 1999 in England.
Sri Lanka have already qualified for the Super Eights with an all-win record in the four-team group.
Former champions India can advance to the next round only if Bangladesh lose to Bermuda, the weakest side in the group who have lost both of their games.
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.
Bangladesh skipper Habibul 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.
India have finished their group campaign with one win and two defeats. Their lone victory came against Bermuda.
Fall of wickets: 1-2 (Outerbridge), 2-8 (Borden), 3-11 (Hemp), 4-35 (Romaine), 5-52 (Pitcher), 6-73 (Cann), 7-86 (Tucker), 8-94 (Minors), 9-94 (Hurdle)
Umpires: Steve Davis (AUS) and Ian Howell (RSA)
TV umpire: Aleem Dar (PAK)
Match referee: Allan Hurst (AUS)