The tourists came into this meeting at Eden Gardens having been hammered by 197 runs in the opening Test, and suffered a huge setback with the news captain and leading batsman Kane Williamson was unable to play due to illness.
However, three early wickets served to raise their spirits before a stand of 141 between Cheteshwar Pujara - whose 87 represented his third fifty from as many innings in the series - and Ajinkya Rahane (77) redressed the balance.
Despite blistering heat in Kolkata, the visiting attack found a second wind and four scalps in the evening session turned the tide back in their favour.
The recalled duo of Matt Henry (3-35) and Jeetan Patel (2-66) proved the most successful bowlers, well backed up by Trent Boult (1-33) and Neil Wagner (1-37), who ended the critical Pujara-Rahane partnership to get his side going again.
Despite the double blow of losing their captain and then the toss, the Black Caps began brightly and had a wicket in the second over as Henry induced Shikhar Dhawan to play on for just one.
Henry removed Dhawan's opening partner Murali Vijay 10 overs later, finding the outside edge with a gem of a delivery as the batsman departed for nine.
And New Zealand were in complete command when India captain Virat Kohli (9), drove at a wide Boult delivery and was smartly held by Tom Latham in the gully.
Following those early losses, Pujara and Rahane set about consolidating in the afternoon session, although it was not without its scares as the former survived a huge lbw appeal off Mitchell Santner while the latter chipped Patel just out of reach of mid-on.
Just as the pair began to accelerate after tea, the partnership was broken as Pujara drove Wagner to short cover, and he was swiftly followed by Rohit Sharma (2), who was well taken at short leg off the bowling of Patel.
And it was Patel who claimed the key wicket of Rahane lbw before the new ball, the introduction of which helped Henry trap Ravichandran Ashwin in front for 26 as New Zealand finished in the ascendancy.