The visitors resumed the fourth day on 24-1 needing 349 to make India bat again, after they were restricted to a first innings score of 193 and India posted a massive 566-8 in reply.
However, the tourists lost seven wickets in the morning session to collapse to 124-8. Pragyan Ojha trapped Brendon McCullum (25) lbw and at the start of his next over bowled Martin Guptil for a first-ball duck, leaving the visitors on 38-3.
Gareth Hopkins (8) went after Suresh Raina took a superb one-handed catch at short leg off the bowling of Harbhajan Singh.
Singh claimed his second wicket of the day in the 29th over after umpire Nigel Long controversially sent Ross Taylor (29) back to the pavilion, despite it appearing that the ball didnÃ¢â‚¬â„¢t hit his bat.
Ishant Sharma bowled Kane Williamson (8) before Raina topped off a great morning in the field by dismissing the injured Jesse Ryder (22) with his second ball and trapping New Zealand captain Daniel Vettori (13) lbw eight balls later, to leave the visitors struggling on 128-8.
India completed the series win when Sharma dismissed Tim Southee (31) and Chris Martin (0) in consecutive balls just after lunch.
The only positive for the tourists, who have struggled throughout this decisive third Test, was the 51-run stand between Southee and debutant Andy McKay (20 not out).
Attention will now turn to the one day series which starts on Sunday.
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BIOGRAPHY: Marcus Rashford