The 30-year-old Gowtham finished with figures of 46.4-10-139-6.
The visitors were 38 for one in 14 overs at the draw of stumps on day three. Ravikumar Samarth (27 not out) and Ankit Bawne (5 not out) were on the crease after Abhimanyu Easwaran was dismissed in the seventh over.
Earlier resuming from 121 for 3, New Zealand managed to take a 75 run first-innings lead, riding on right-hand batsman Cameron Fletcher's career-best innings of 103.
Fletcher hit seven boundaries and a six in his 221-balls innings.
Gowtham drew first blood of the day by dismissing overnight batsman Rachin Ravindra. Other overnight batsman Seifert departed next, following which Bracewell joined Fletcher at the crease.
Bracewell's innings was studded with nine boundaries. He was dismissed by Navdeep Saini with the host at 257 for six.
The duo of Fletcher and Jamieson then put up a 100 run stand which ended with Gowtham dismissing the latter.
India A: 323 and 38/1 in 14 overs (R Samarth 27 not out; D Bracewell 1/17)
New Zealand: 398 all out in 131.4 overs ( C Fletcher 103, T Seifert 86; K Gowtham 6/139).
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