Rohit (76 not out) and fellow opener Shikhar Dhawan (31 not out) made light work of England's 110, knocking off the victory target with 31.2 overs to spare.
Earlier, England had a horror start after being put in to bat when Bumrah rattled Jason Roy's stumps with an in-swinger and then deceived Joe Root two balls later with a delivery that nipped away. Both players gone for ducks.
New Test captain, Ben Stokes, also perished for nought in the following over, edging behind off Mohammed Shami (3-31) first ball, before Bumrah accounted for Jonny Bairstow (7) and Liam Livingstone (0) to leave England reeling at 26-5.
"Looking at the overhead conditions and the pitch, the toss was a right call to make. We used the conditions upfront really well," said Rohit.
"There was some swing and seam upfront and we exploited them well."
Newly appointed captain Jos Buttler (30) and Moeen Ali (14) led a brief recovery but both were dismissed in quick succession before a 35-run ninth-wicket stand between David Willey (21) and Brydon Carse (15) pushed England past 100.
The home side were eventually bowled out in 25.2 overs for their lowest total against India in 50-over cricket. Their previous lowest - 125 - came at Jaipur in 2006.
"We have some guys in the form of their lives in Tests coming here and getting nicked off," said Buttler. "Jasprit is a great bowler to come up against. He has bowled fantastically well today and deserves these figures."
"I won't be panicking... in the last seven years our batting has been our super strength in ODI cricket and the guys in the teams are some of the best players to ever play the format for England so we don't want to panic."
The second match of the series takes place at Lord's on Thursday. Virat Kohli could be a doubt for that encounter having missed India's scintilating victory with a groin strain.
