The visitors had finished their first innings on day two at 364, after bowling out the hosts for 180 on the first day.
Shardul Thakur (0-11) and Ravindra Jadeja (1-25) started the day's proceedings, and the Mumbai pacer got some movement immediately.
Thakur was on the mark with his line and length, just as in the first innings and troubled the batsmen. However, unlike day one, he wasn't able to garner any success in the five overs he bowled.
Opener John Campbell stepped out and smacked him for two sixes, and thereafter Ravichandran Ashwin (2-38) was re-introduced into the attack.
The left-arm spinner didn't get any rest though, as skipper Virat Kohli deployed him from the same end as in the first innings. And it paid immediate dividends as Jadeja got his man, with Campbell caught at mid-on by KL Rahul.
In the end, the wicket did come off a brilliant turning ball, with Hope caught by Kohli at leg slip in the 26th over of the innings.
Mohd. Shami (0-17) returned thereafter to bowl a four-over spell, while Amit Mishra (0-11) was deployed as well. But Blackwood, in company of Vishaul Singh (17 batting), managed to bat on for nearly 17 overs and put on 34 runs for the fourth wicket going into the break.
Besides Jadeja, KL Rahul hit 64 off 127 balls and Kohli made 51 off 94 balls, as the visitors ended their first innings in 105.4 overs.
Jadeja hit a superb 61-ball 56 down the order with the help of eight boundaries.
(REOPENS DEl 77)
The only blemish in Kohli's near flawless innings was a dropped chance by Southee off his own bowling in the 28th over, that too after smacking the pacer for boundaries of consecutive balls.
But by then the job was almost done and Kohli in Kedar Jadhav's (10 not out) company brought up the winning runs to pile misery on the Kiwis, who are yet to taste a victory in the tour.
Earlier, Pandya (3/31), who was handed over his ODI cap by legendary Kapil Dev, opened the attack along side comeback man Umesh Yadav (2/31) as the duo wreaked havoc in the initial overs to rattle the Black Caps innings.
Opting to bat after winning the toss, India witnessed a dream start as they reduced New Zealand to 43 for four in 10.4 overs.
Debutant Pandya gave India the first breakthrough in the final ball of the second over when he removed out-of-form Martin Guptill (12).
Struggling for runs throughout the tour, Guptill hit Pandya for three boundaries in the same over before the right-arm medium pacer took his revenge, inducing an outside edge as the batsman jabbed at a length delivery and Rohit Sharma did the honours at the second slip.
It was a gem of a delivery from Umesh, which pitched on good length outside off and moved slightly to get an outside edge and Dhoni did the rest behind the stumps.
Returning to action after change of ends, Pandya dismissed Corey Anderson (4), but the credit for the wicket should go to Umesh, who pulled off a superb catch at mid-off.
An over later, Pandya picked up his third wicket of the day in the form of Luke Ronchi, who flicked one straight to Umesh at mid-on as New Zealand slumped to 49 for five at the end of the 13th over.
Renshaw and David Warner played with a positive intent and cruised to 50 for no loss in 9.3 overs.
But after spin was introduced at both ends in the 10th over, Ravindra Jadeja (1/49) struck in his first over with David Warner (19) offering a simple return catch to the left-arm spinner.
Renshaw's first 24 runs came only in boundaries as once he cleared a fielder the ball would race past the rope.
Seventh over and Aswhin came in with his bunny' Warner at the opposite end but without any immediate effect.
With Warner looking to break free, Kohli employed spin attack at both ends and Jadeja was welcomed by the Aussie vice-captain with another crunching drive.
The next ball, a juicy full-toss and Warner offered Jadeja a return catch for the first breakthrough for India.
In the middle of a fine run, 20-year-old Renshaw looked to complete a third half-century in as many Tests, before falling six runs short.
Yadav produced the breakthrough before lunch in his second spell when Renshaw guided to Kohli at first slip for a soft dismissal.
