Left-handed Anderson smashed 70 off 68 and balls and took a stunning catch, but India held their nerves amid mounting tension to squeak home by three wickets in a rain-hit semi-final under lights.
Sreevats Goswami made 51 and captain Virat Kohli hit a fluent 43 as India, set a revised target of 191 from 43 overs, survived a middle-order collapse to win with nine deliveries to spare.
New Zealand had made 205-8 after electing to take first strike in overcast conditions at the Kinrara Oval.
India await the winners of Friday's second semi-final between defending champions Pakistan and South Africa in the title clash on Sunday.
The Indians lost in-form opener Taruwar Kohli in the seventh over and star batsman Tanmay Srivastava in the 14th to slip to 40-2.
Srivastava fell to a remarkable catch at mid-on by Anderson, who dived to his left and plucked the ball in mid air after the batsman had played an uppish drive off India-born Anurag Varma.
Left-hander Goswami and Kohli steadied India with a 84-run stand for the third wicket, but India lost four wickets for 33 runs to almost mess up the chase.
India, cruising at 150-3 at one stage, dipped to 183-7 before Saurabh Tiwary took his team home with an unbeaten 29 which included the winning two runs off spinner George Worker.
Earlier, New Zealand lost their first two wickets for 29 runs by the 12th over.
Skipper Kane Williamson (37) and Fraser Colson (32) retreived the situation by adding 57 for the third wicket when Virat Kohli removed both batsmen in successive overs.
Kohli, who alternated between seam bowling and off-spin, had Williamson stumped smartly by wicket-keeper Goswami off a delivery deliberately bowled down the leg-side.
In his next over, Kohli shattered Fraser's stump with a ball that kept low and left New Zealand tottering at 90-4 in the 29th over.
Anderson, a 17-year-old who plays first-class cricket for Canterbury, took charge and swung Kohli and left-arm spinner Iqbal Abdullah for sixes early in his innings.
Tim Southee (11) helped Anderson add 47 for the fifth wicket to carry the total to 137-4 when seamer Siddarth Kaul struck twice in successive balls to halt the New Zealand charge.
Anderson continued to lash out, slamming a six and four in one over from Kaul and meted out the same treatment to left-armer Srivastava soon after.
Srivastava avenged the hammering by removing Anderson in the 49th over to end a seventh-wicket stand of 58 with Michael Bracewell.
Bracewell, nephew of former Test spinner and current Black Caps coach John, remained unbeaten on 17.
Kaul, Kohli and Srivastava picked up two wickets each for the Indians.
New Zealand 205/8 in 50 overs (Anderson 70, Ken Williamson 37, Colson 32; Virat Kohli 2 for 27, Siddarth Kaul 2 for 43, Tanmay Srivastava 2 for 46) lost to India 191/7 in 41.3 overs (Sreevats Goswami 51, Virat Kohli 43, Saurabh Tiwary 29 n o; Tim Southee 4 for 29, Boult 1 for 26) by three wickets (D/L method).