Dhoni (50) and Jadeja (77 in 59 balls) gave India hope by putting on a World Cup-record 116 for the seventh wicket but they were both dismissed, leaving the team needing 23 off the final over.
Kohli suggested the tournament follow the model of the Indian Premier League, in which teams who finish top in the regular season get another chance to reach the final if they lose their first play-off match.
The Black Caps will play hosts England or holders Australia in Sunday's title match at Lord's after pulling off a remarkable triumph against two-time champions India. New Zealand will meet the victor of the second semifinal at Lord's on July 14.
India KL Rahul, R Sharma, V Kohli, R Pant, MS Dhoni, D Karthik, H Pandya, R Jadeja, B Kumar, J Bumrah, Y Chahal. If Australia wins, it will be a repeat of the 2015 final won by the Australians in Melbourne.
It would be the 33rd over before they finally broke into three figures, MS Dhoni coming well down the track to strike the first six of India's innings.
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The 40-year-old Tendulkar, who is now serving as a television pundit for Star Sports the official broadcaster of the World Cup in India, said in an interview with India Today: "I have said in a number of interviews that Kane Williamson as captain has a smart, thinking brain and he is always in control".
"We played outstanding cricket throughout this tournament, and to just go out on the basis of 45 minutes of bad cricket is saddening and it breaks your heart".
New Zealand scored 28 runs after the resumption under cloudy but brightening skies, finishing on 239-8.
For the first time in global history, a team lost its first three batsmen all for 1 as India saw Sharma, Kohli then Rahul depart in the space of 10 balls and was marooned on 5-3 after 3.1 overs.
Kohli only lasted six balls, before he was trapped lbw by Trent Boult.
The India captain reviewed, indicating he thought the ball was going over the top.
The top-order collapse saw India stranded at 24/4 when Rishabh Pant and Hardik Pandya joined hands to weave a 47-run partnership for the fifth wicket before a slog sweep off Mitchell Santner saw Pant (32 off 56 balls) walk back to the pavilion, much to the dismay of skipper Virat Kohli in the dressing room.
That brought Dhoni and Jadeja together and they almost saw India home in an improbably nerve-racking finish in Manchester.