Paceman Rahman claimed five for 50 as the hosts posted their highest total against India before destroying their neighbours' vaunted batting lineup in 46 overs in front of a boisterous crowd at Mirpur's Shere Bangla National Stadium.
Chasing 308 to win, India got off to a strong start before the wheels came off their chase with only Rohit Sharma (63) making a meaningful contribution as their top order capitulated.
"I cannot ask for more, I'm really happy with the performance from the boys," an elated Bangladesh captain Mashrafe Mortaza said.
Opting to bat first, Bangladesh were given a flying start from openers Tamim Iqbal (60) and Soumya Sarkar (54).
All-rounder Shakib Al Hasan (52) also reached his half century as Bangladesh posted a total of 307 and were all out with two balls to spare.
Off-spinner Ravichandran Ashwin (3-51) was the pick of the Indian bowlers.
Shakib claimed the important wicket of India skipper Mahendra Singh Dhoni even though it was Taskin Ahmed's double strike that pegged back India.
Shikhar Dhawan (30) and Sharma had added 95 runs to give India a strong start when Ahmed dismissed the former and sent back Virat Kohli in his next over.
Rahman, 19, then wrecked India's top order by dismissing Sharma and Ajinkya Rahane and went on to claim three more wickets before India folded for 228.
Earlier, Tamim and Sarkar launched a blistering attack on the Indian bowlers during a 102-run opening stand before being separated in the 14th over.
Sarkar set out for a non-existent single before being sent back by Tamim but the batsman could not beat Suresh Raina's direct throw.
When play resumed after a rain-interruption, Tamim, who had hit Umesh Yadav for three boundaries and a six in the same over, was dismissed by Ashwin who took three quick wickets to trigger a batting collapse.
Shakib then steadied the innings while the lower order took the hosts past the 300-mark.
"Bangladesh played some very good cricket, from very first ball they took on our bowlers. They were brave enough to play the big shots and it paid off," Dhoni said.
The teams return to the same venue for the second ODI on Sunday.
(Reporting by Amlan Chakraborty in New Delhi; editing by Pritha Sarkar)
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