Ashwin, who claimed 6-81 in the first innings, resumed his role as New Zealand's wrecker-in-chief, claiming 7-59 as the tourists, chasing 475 for an improbable win, were dismissed for 153 to lose inside four days.
After Umesh Yadav dismissed Tom Latham to earn an early breakthrough, Ashwin spun a web around the batsmen as New Zealand lost nine wickets in the final session.
Ashwin dismissed captain Kane Williamson (27) for the second time in the match, ended Ross Taylor's breezy 32 and sent back Luke Ronchi to rip out the Kiwi top order before returning to blow away the tail.
Fittingly, Ashwin caught Trent Boult off his own bowling to end New Zealand's innings with career-best bowling figures.
The spinner finished the series with 27 wickets and was the obvious choice for the man-of-the-match and man-of-the-series awards.
"Could not have asked for anything better," India captain Virat Kohli said at the presentation ceremony.
"The bowlers have responded beautifully in the series every time we've been under a bit of pressure. Every time the Kiwis had a partnership, someone put his hands up and brought those breakthroughs for us."
Earlier, Gautam Gambhir struck 50 on his test return and Pujara scored 101 not out as India declared their second innings on 216 for three around an hour before tea.
Resuming on 18-0, the hosts lost opener Murali Vijay cheaply but Gambhir, who had retired hurt with an injured shoulder on Monday, returned to score fluently.
Gambhir hit six boundaries before completing his 22nd test fifty off 54 balls but was dismissed when he drove off-spinner Jeetan Patel to Guptill at cover.
Patel removed Kohli (17) lbw for the second time in the match but Pujara and Ajinkya Rahane (23 not out) added 58 quick runs before India declared.
"It was important that we get a big total in the first innings but it was not to be," Williamson said.
"Credit to India, obviously they're the number one in the world now. A fantastic effort and well-deserved."
(Reporting by Amlan Chakraborty in New Delhi; Editing by Ed Osmond)
