Mustafizur Rahman took 6-43 for Bangladesh as they bowled India out for 200, with opener Shikhar Dhawan top-scoring with 53 off 60 balls.
In reply, Shakib Al Hasan made 51 not out as Bangladesh comfortably won with nine overs to spare.
Bangladesh beat India by 79 runs in the first ODI on Thursday.
The result caps a remarkable period for the side who knocked out England of the World Cup in March. Bangladesh beat the English by 15 runs in Adelaide, with coach Peter Moores being sacked two months later.
Despite one game still left to play, Bangladesh clinched their first ever series win over India, aided by pace bowler Mustafizur's 11 wickets in the two matches so far.

Performers dressed as cartoon characters popular in the west were seen dancing on state TV, along with violin players in miniskirts.
Pyongyang has long taken pains to keep popular culture out of its media.
This seems to be the first time that Disney characters have been featured in the reclusive state.
A Disney spokesperson told the Associated Press news agency that it had not licensed or authorised the use of the characters in the show.
'Grandiose plan'
The concert was staged by the Moranbong band as its debut performance on Friday, reported state media KCNA.
The band was put together by Kim Jong-un, who has a ''grandiose plan to bring about a dramatic turn in the field of literature and arts this year", state media said.
North Korean entertainment has traditionally featured nationalist or folk songs, and mass displays.
In recent years, reports say, children in the isolated country have become familiar with characters such as Mickey Mouse and Winnie the Pooh through stationery items and clothes imported from China, North Korea's closest ally.
It seems to point to an easing of North Korea's paranoia about what it calls spiritual pollution from the West, and even more dangerously, South Korea, says the BBC's Charles Scanlon.
It is still illegal to watch South Korean television programmes in North Korea. But a recent report based on interviews with refugees said that North Koreans now have more access to outside media, especially DVDs of popular South Korean drama serials.
The young leader, who took power following the death of his father, Kim Jong-il in December 2011, has been seen as attempting to make changes to the profile of the North Korean leadership.
Mr Kim, believed to be in his late 20s, made his first public speech in April, in a 20-minute televised address as Pyongyang marks the centenary of the birth of the country's founder, his grandfather, Kim Il-sung.
Kim Jong-il is thought to have made one recorded public utterance in his 18-year rule - and that was a single sentence.
Bangladesh beat India by six wickets on the Duckworth-Lewis method to take an unassailable 2-0 lead in their one-day international series.
[NEXT_CONCEPT]
Mickey Mouse and other Disney characters have shown up on stage in North Korea in a concert for leader Kim Jong-un, raising some eyebrows.