The hosts trail India 2-1 and can no longer win the five-match series, but they can avoid defeat if they win the final match at the same venue on Sunday. The third match was abandoned due to rain.
Chasing 334 for victory, India got off to a disastrous start as paceman Henry ripped through their top order in a lethal first over, taking the wickets of Rohit Sharma, Virat Kohli and Shreyas Iyer to leave the visitors reeling on 0-3.
Ruturaj Gaikwad also departed shortly afterwards, but a steely 44-run partnership between Ishan Kishan (27) and Axar Patel (21), and a quickfire 21 from Naman Dhir helped India avoid a record defeat by the slimmest of margins.
Their worst-ever defeat in terms of runs came in a 245-run loss to Sri Lanka in 2000.
Henry took three more wickets as India were bundled out for their sixth-lowest ODI total.
"With the way the wind was going, I wasn't too worried about the swing," said Henry, who finished with 6-27.
"From ball one it was just about trying to have the strengths and bring the stumps in play, really. We knew that there was a small window there with the new ball that you could put the batters under pressure.
"So just trying to be as simple as possible, obviously, and when it moves sharply like that in the first over, when people are facing first balls, pretty tough," he said.
Earlier on Friday, New Zealand opener Young made up for his single-digit scores of nine and six in the opening two matches of the series with an assured knock of 125 from 108 balls - his best-ever ODI score and his fifth century in the format.
The 33-year-old profited from some lax fielding by India, with Rohit and Kishan dropping him on consecutive deliveries in the second over of New Zealand's innings.
Young was supported by Tim Seifert (58) and Finn Allen (41), while India's Patel, Gurnoor Brar and Auqib Nabi each picked up two wickets.
India were without captain Shubman Gill and batter KL Rahul, who will also miss the final ODI as they prepare for the upcoming two-Test series against New Zealand beginning on Wednesday.
"There was a huge difference in the first innings and the second innings in terms of the wicket, but also credit where it's due, the way Henry bowled in the right areas right from ball one," stand-in captain Iyer said.
"It's important to forget about this one, obviously, analyse what went wrong in terms of bowling and batting. But at the back of our minds, it's important that we have also won two games."