New Zealand, following on after conceding a 279-run lead, were bowled out for 164 in their second innings in the final session.
Off-spinner Ravichandran Ashwin took six wickets and left-arm spinner Pragyan Ojha three for India, while Kane Williamson top-scored for New Zealand with 52.
The second and final Test starts in Bangalore on August 31.
Brief scores:
India 438; New Zealand 159 and 164.