The 103-run stand for the third wicket finally came to an end in the 24th over as Goswami dismissed Satterthwaite (59), with the White Ferns still 163 runs away from the target.
Sneh Rana got the crucial wicket of Amelia Kerr (67) in the 31st over, reducing New Zealand to 152/4, with the hosts still 128 runs away from the target. Renuka Singh then got her maiden ODI wicket in the 34th over as she sent Maddy Green (24) back to the hut and this left the hosts in a spot of bother.
Lauren Down and Katey Martin got together at the crease and the duo led White Ferns' fightback in the contest. Both batters stitched together a partnership of 76 runs for the seventh wicket but with New Zealand 33 runs away from the target, the hosts lost the wicket of Martin (35). Down brought up her half-century in the 47th over of the innings and in the last three overs, New Zealand needed 23 runs to win.
In the end, Down (64*) and Frances Mackay (17*) ensured that New Zealand took a 3-0 lead in the series.
Earlier, Deepti Sharma played an unbeaten knock of 69 while Sabbhineni Meghana (61) and Shafali Verma (51) also chipped in as India posted 279 runs on the board. On a rare-off day, Mithali Raj failed as she was dismissed after scoring just 23.
Harmanpreet Kaur's poor form continued as she was sent back to the pavilion after scoring just 13 off 22 balls. For New Zealand, Hannah Rowe and Rosemary Mair returned with two wickets each.