Ashwin strikes as New Zealand 92-2 at lunch in second India Test
New Zealand won the toss and elected to bat on a pitch that is expected to keep the spinners in business in Pune.
Devon Conway, on 47, and fellow left-hander Rachin Ravindra, on five, were batting at the break after the vistors lost Latham and Will Young.
The Black Caps won the opener for their first Test victory in India since 1988 and are looking to clinch the three-match series.
The eight-wicket win in Bengaluru was New Zealand's third Test victory in India, but they have never won a series in the country.
Latham and fellow left-hander Devon Conway started cautiously, but soon got going with a few boundaries to blunt the opening pace attack of Jasprit Bumrah and Akash Deep.
Spin was introduced in the eighth over and Ashwin struck on his fifth ball as captain Latham was trapped lbw for 15 to leave New Zealand 32-1.
Conway then put on 44 runs with Young and the two looked comfortable -- before Ashwin struck again, to get Young caught behind for 18.
Initially given not out, India reviewed and technology showed Young got a tickle off the glove while trying to work the ball down the leg-side.
New Zealand have bolstered their spin force with left-arm orthodox Mitchell Santner replacing injured fast bowler Matt Henry.
India made three changes from their opening defeat with fit-again batsman Shubman Gill, Deep and spinner Washington Sundar coming into the team.
Sarfaraz Khan kept his place after his 150 in the first Test with struggling batsman KL Rahul making way for Gill, who has recovered from a neck injury.