Resuming the game from 129/0, Latham and Young got off to a cautious start as India's hunt for wickets continued.
Ashwin got the much-needed breakthrough for India as Young departed after scoring 89 runs in the 67th over.
Williamson and Latham then continued New Zealand's innings to pile runs against India in the first innings before Umesh removed the Kiwi skipper.
Earlier on Friday, New Zealand opening batters Will Young and Tom Latham registered half-centuries as the Kiwis dominated the second day of the Test match.
After bundling down India for 345 in the first innings, the visitors dominated the proceedings as the hosts looked for wickets in the third session on Friday.
Brief Scores: India 345/10 (Shreyas Iyer 105, Ravindra Jadeja 50; Tim Southee 5-69) vs New Zealand 197/2 (Will Young 89, Tom Latham 82; Umesh Yadav 1-30)