The ground was only about two-thirds full at the start of play on the reserve day, which was required after persistent rain brought a premature end to play Tuesday.
The victor will play England or Australia in final at 9.30pm Sunday.
Kohli lauded India's show overall in the World Cup, only pointing to stray instances of poor shot selection as areas for improvement.
Rohit Sharma fell for just one against New Zealand but had played a tremendous tournament until that point to advance his claim to be the best ODI opener in the world.
India's top order crumbled to 4-24 after 10 overs as three wickets to Henry, plus the crucial scalp of Virat Kohli claimed by Boult, left the top ranked side reeling.
But a superb display with the ball from Matt Henry (3-37) and Trent Boult (2-42) helped bowl their opponents out for 221, the 48th-over dismissal of Ravindra Jadeja after an eye-catching 77 proving pivotal.
Resuming at 211 for 5, the Black Caps could add only 28 runs to their overnight tally with a single boundary being hit off the remaining 23 balls. Rishabh Pant was the top scorer with 32 runs, but he too was dismissed by Mitchell Santner in the 22nd over with the required run rate quickly rising for the Asians.
India needed 31 off the final two overs.
Williamson had won the toss on Tuesday and opted to bat at a venue where teams batting first had won all five previous matches in the tournament. "I don't think India need to make any drastic changes, they just need to look at that middle-order", he said.
However, in this instance, Jadeja produced one of the best innings in recent memory and transformed his side's fortunes with some spectacular pyrotechnics as New Zealand began to feel the rug being pulled from under their feet.
Faced with a seemingly unattainable target, the Indians sensed urgency, and Dhoni and partner Bhuvneshwar Kumar pushed for a second run on the fourth ball of the penultimate over.
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