Ashwin ended the first Test with 10 wickets (four in the first innings, six in second innings) as India ran out deserved winners to take a 1-0 lead in the three-match series.
Ashwin, who had claimed the first three New Zealand wickets to fall on Sunday, returned to haunt the Kiwis on the final day of a competitive Test match.
Ravindra Jadeja, who was adjudged the man-of-the-match, dismissed Luke Ronchi (80) to give India the first breakthrough of the day.
Mohammed Shami took two wickets in two balls as India edged closer to win.
Ashwin then took it upon himself, first ending Mitchell Santner's (71) gritty vigil. He then uprooted Ish Sodhi's leg stump to claim his 19th five-wicket haul.
The off-spinner put the final nail in New Zealand's coffin by catching Neil Wagner plumb in front of the stump to give India the victory.
So, Ashwin began the Kiwi downfall by taking the first three wickets of the second innings, then ended their resistance by claiming the final three.
Ravichandran Ashwin Joins Elite Club
On Sunday, Ashwin joined the 200-wicket club when he dismissed New Zealand captain Kane Williamson on Day 4. Ravichandran Ashwin, thus, became the fastest Indian and second fastest overall to take 200 wickets in Test cricket.
Ashwin reached the feat in his 37th match while the next best is Harbhajan Singh, who took 46 matches to join the 200 club.
India's current head coach Anil Kumble is the third fastest, taking 47 matches while another legendary leg spinner BS Chandrasekhar (48) is fourth in the list.
Kapil Dev (50 matches) is the fifth in the list for India but is the fastest non-spinner to reach the feat.
Australian spinner Clarrie Grimmett is the fastest bowler overall to take 200 wickets. Another Australian Dennis Lillee along with Pakistan pacer Waqar Younis are joint-third fastest having reached the landmark in 38 Tests.
Ashwin is only the ninth Indian to take 200 or more wickets. Kumble is the leading wicket-taker for India with 619 scalps to his name.