At stumps, the visitors were 106 for five, trailing India by 332 runs in their first innings. James Franklin (31) and Kruger van Wyk (0) were at the crease for New Zealand. Off-spinner Ashwin took three wickets while left-arm orthodox spinner Ojha picked up the remaining two to fall after their batting colleagues racked up 438 in their first innings.
"We should be eyeing a follow-on. With respect to how the wicket is behaving and the weather being a little cloudy, we should be looking to bowl twice and get the job done," Ashwin said.
"If you put 440 on the board you expect yourself to be in a dominant position in the game. It's a good enough score to dominate the game."
Ojha, who was introduced in the eighth over, struck first when he lured opener Brendon McCullum (22) to drive straight into the hands of Virat Kohli at cover.
Kohli took the next two catches at leg slip off Ashwin to dismiss Martin Guptill (2) and New Zealand captain Ross Taylor (2).
Daniel Flynn (16) was adjudged leg before off Ashwin, which reduced New Zealand to 55 for four.
Kane Williamson (32) and left-handed Franklin provided some resistance with a 44-run stand for the fifth wicket before Williamson became the last wicket to fall in the day.
"The new ball was a little bit hard and the seam was much more upright, something that helped us extract bounce and enabled us get a few wickets up front," Ashwin said.
"As the ball gets old, it would slow up a little bit and we have to be more patient to get wickets tomorrow."
Earlier, Cheteshwar Pujara (159) and MS Dhoni (73) added 127 runs for the sixth wicket for India before the visitors polished off the tail.
Pujara and Dhoni both fell trying to attack off-spinner Jeetan Patel, who finished with four for 100. Left-arm paceman Trent Boult chipped in with figures of three for 93.
Resuming on their overnight score of 307 for five, Pujara and Dhoni enjoyed a trouble-free morning session, which was delayed for 25 minutes due to the threat of rain.
Batting at the No 3 spot vacated by the recently retired Rahul Dravid, the 24-year-old Pujara continued to display the cool temperament that took him to a maiden Test century in his fourth appearance for India.
Showing little sign of rustiness in his first Test in more than 18 months, he finally fell when trying to smash Patel down the ground but only succeeded in picking out Franklin, who pouched a well-judged catch at mid-on.
Pujara hit 19 boundaries and a six during his 306-ball stay at the crease.
Dhoni, the India captain, who hit six fours and a six, fell in similar fashion trying to plunder quick runs from the off-spinner, only to be caught out mid-off.
Ashwin contributed with a cameo of 37, including five boundaries.
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