Sent into bat, Indian batsmen produced a below-par performance to be bowled out for a paltry 105 runs in rain-truncated 41 overs-a-side game.
But the pace combination of Binny (6-4) and Mohit Sharma (4-22) saved India from humiliation as they bundled out Bangladesh for a record 58 runs in 17.4 overs to help the visitors defend their lowest score against the hosts.
The Binny heroics came after Bangladesh pacer Taskin Ahmed scalped five wickets for 28 runs on his debut to bowl India out for 105 in 25.3 overs.
But if India's 105 was much less than their previous low of 191 in Port of Spain during the 2007 World Cup then Bangladesh also sank to a new low as their previous lowest against the Men in Blue was 76 in 2003 at Dhaka.
Binny, whose figures are best by any Indian bowler in ODI's bettering Anil Kumble's 6-12 against West Indies in 1993, was ably supported by medium pacer Mohit from the other end.
Mohit, who was the proud owner of the 'Purple Cap' in the recently concluded IPL for Chennai Super Kings, started the rot for Bangladesh after sending openers Tamim Iqbal for 4 and Anamul Haque for nought.
Binny soon joined forces to send innings top-scorer Mithun Ali back in the pavilion for 26 and skipper Mushfiqur Rahim for 11.
The rest of the batting fell like a pack of cards as Binny and Mohit had little time to celebrate between wickets.
Earlier put into bat, the stunning Indian collapse was triggered by Taskin, who returned with figures of 5-28, making it the best debut bowling performance by a Bangladesh player.
Skipper Suresh Raina's 23-ball 27, laced with three fours, was the best that an Indian batsman could manage under overcast conditions.
Put into bat, India lost Ajinkya Rahane for nought off Mashrafe Mortaza but rain soon halted play after 5.1 overs.
The resumption of play after three hours did not help matters for the visitors as Robin Uthappa (14) top-edged a skier to give Taskin, 19, his debut wicket in overcast conditions.
The lanky pacer continued to trouble the inexperienced Indian batsmen in conditions conducive to swing bowling.
Cheteshwar Pujara (11) and Ambati Rayudu (1) were trapped LBW by Taskin.
Raina tried to counter-attack with three boundaries but an Al-Amin Hossain-Mortaza combination ran him out.
Wicketkeeper-batsman Wriddhiman Saha (5) was soon trapped LBW off Mortaza and Akshar Patel was cleaned up by Al-Amin as India looked straight down the barrel.
All-rounder Binny (3) was caught behind chasing a wide delivery from Taskin.
The procession back to the pavilion continued as Amit Mishra was trapped LBW and Taskin's joy knew no bounds after he picked up his fifth wicket and became the eighth bowler in the history of ODIs to bag a fifer in his maiden match.
Taskin also became the first Bangladesh bowler to bag five wickets on debut. The previous best by a Bangladeshi was achieved by Sohag Gazi and Rubel Hossain, both of whom scalped four wickets on debut.
The Men in Blue laboured hard to cross the three figure mark but the formalities were completed after Shakib Al Hasan scalped Umesh Yadav (17) to bring an end to India's misery.
In fact, had it not been for the 13-ball cameo by Yadav, India could well have folded inside the 100-run mark.
The final match of the series is slated for June 18 at the same venue.