Cheteshwar Pujara had set up the victory bid in the third Test in Indore when he hit an unbeaten century as the hosts set the Black Caps 475 to win.
Off-spinner Ashwin then returned his best innings figures of 7-59 as New Zealand were skittled for 153.
Virat Kohli (211) and Ajinkya Rahane (188) helped India to a decisive 557-5 in the first innings.
Their partnership of 365 was India's highest for the fourth wicket, and Ashwin took 6-81 to restrict New Zealand to 299 in their reply.
Ashwin, 30, finished with 27 wickets in the three-Test series.
The whitewash series win solidifies India's position at number one in the Test rankings, which they won back from Pakistan after the second Test in Kolkata.
Ravichandran Ashwin took a career-best 13-140 as India beat New Zealand by 321 runs to seal a 3-0 series whitewash.