Rahman, 19, who claimed five for 50 in Bangladesh’s 79-run win on Thursday at the Sher-e-Bangla National Cricket Stadium in Dhaka, followed up with six for 43 at the same venue on Sunday to help clinch his side a historic six-wicket win.
This latest triumph for Bangladesh over their mighty foe was only their fifth ever and gave them an unassailable 2-0 lead in the three-match series. It also cemented their place in the 2017 Champions Trophy.
Zimbabwean Brian Vitori is the only other player to achieve successive five-wicket hauls in his first two one day internationals – against Bangladesh in 2011.
India, skittled out for 228 on Thursday, elected to bat after winning the toss and were eventually bowled out for 200 in 45 overs after rain had reduced the game to a 47-over match.
Rahman set the trend by taking the wicket of Indian opener Rohit Sharma for a duck with the second ball of the contest and went on to snare dangerman MS Dhoni (47), Suresh Raina (34) and Ravindra Jadeja (19).
He then trapped Axar Patel lbw first ball to set up a hat-trick in front of a delirious home crowd and although that was missed he removed Ravichandran Ashwin (four) soon after.
Opener Shikhar Dhawan, who struck 53 off 60 balls, was the only India batsman to make a half-century as the hosts were set a Duckworth-Lewis revised victory target of 200 from 47 overs.
Bangladesh set about their modest run chase at a steady pace and when opener Tamim Iqbal was caught by Dhawan off Dhawal Kulkarni for 13 in the seventh over they had 34 runs on the board.
Soumya Sarkar (34) and wicketkeeper Litton Das (36) lifted Bangladesh to 86 in the 17th over.
But Sarkar fell to Ashwin lbw and when Das was caught by Dhoni behind off Patel 12 runs later India had the hosts at 98 for three in the 20th over and were still in the game.
Bangladesh still needed 48 when Mushfiqur Rahim (31) was brilliantly run out by Dhoni, but star man Shakib Al Hasan dispelled any doubts with a commanding 51 not out off 62 balls to ease the hosts home.
Al Hasan put on an unbeaten 48 with Sabbir Rahman (22no) as Bangladesh reached their target with six wickets and 54 balls to spare to spark scenes of wild jubilation at the national stadium.