Saha smashed an unbeaten knock of 61, and was involved in an unbeaten 67-run partnership with Axar Patel as India declared their second innings at 234 for seven.
Earlier, India had put 345 runs in the first innings thanks to Iyer's maiden Test ton on his debut appearance in the longest format of the game.
In reply, New Zealand couldn't capitalise on the brilliant start given by openers Tom Latham (95) and Will Young (89) as the Kiwis were bowled out for 296, thanks to Axar Patel's five-for.
The Black Caps now need 284 runs to win the first Test and take a 1-0 lead in the two-match Test series.
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