Put in to bat, India were all out for 329 in 48.2 overs with Shikhar Dhawan (67), Rishabh Pant (78) and Hardik Pandya (64) hitting half centuries.
Curran (95 not out off 83 balls) India single-handedly kept England in the hunt as he shepherded number 10 Mark Wood (14) for a 60-run stand for the ninth wicket but in the end could not score 13 runs needed in the final over.
England ended at 322 for 9 in 50 overs.
India dropped at least four catches, including that of Curran, at crucial junctures as they were kept frustrated till the end.
The Indian innings was built on two big partnerships -- 103 for the opening wicket between Rohit Sharma (37) and Dhawan and 99 for the fifth wicket between Pant and Pandya.
Pant's sparkling knock came off 62 balls while Pandya needed 44 for his blistering innings but India lost their last four wickets for a mere eight runs.
Pant, during his career-best knock, plundered four maximums and five boundaries, while pandya smashed five fours and four sixes.
For England, Mark Wood was the most successful bowler with figures of 3 for 34 while Adil Rashid took two wickets.
Chasing the target, England lost wickets at regular intervals and could not stitch any substantial partnership on a flat pitch before Curran's heroic effort.
Shardul Thakur returned with figures of 4/67 while Bhuvneshwar Kumar got three.
India had won the preceding four-match Test series 3-1 and the five-match T20 International contest 3-2.
India: 329 all out in 48.2 overs (Rishabh Pant 78, Shikhar Dhawan 67, Hardik Pandya 64; Mark Wood 3/34).
England: 322 for 9 in 50 overs (Sam Curran 95 not out; Shardul Thakur 4/67, Bhuvneshwar Kumar 3/42).