The hosts were 75 for three in the second innings, 166 runs behind India who were bowled out just before the tea break for 520 in reply to New Zealand's 279.
New Zealand made a disastrous start to their second innings with opener Tim McIntosh controversially caught at first slip by Tendulkar off the third ball for a duck.
Television replays suggested the ball may have bounced just in front of Tendulkar before he closed his hands around the ball. The 35-year-old Tendulkar immediately left the field to receive treatment to his hand.
New Zealand coach Andy Moles said he thought the catch should have been referred to the third umpire.
"You all saw the TV shots, it's fair to say we're surprised it wasn't referred," Moles told reporters.
Tendulkar, however, disagreed.
"I have the seen replays and I have seen my fingers under the ball," he said.
"Sometimes on camera it looks different. I was pretty much confident otherwise I would not have appealed for it."
Debutant Martin Guptill (48) and Daniel Flynn combined for a 68-run second wicket stand before Guptill threw his wicket away in sight of his first test half century when he smashed a Harbhajan Singh half volley to Virender Sehwag.
New Zealand's crisis deepened when night watchman Kyle Mills, who lasted 19 balls in the fading light, was trapped in front by Munaf Patel for two on the last ball of the day.
Flynn was left on 24 not out and will be joined at the crease on Saturday by Ross Taylor with New Zealand's last three specialist batsmen in Flynn, Taylor and Jesse Ryder needing to bat the entire day to avoid a heavy loss within four days.
"We find ourselves in a very difficult position," Moles said.
"We've been outplayed for the first three days and we've got to make sure we show a bit more character over the next two.
"We've under-performed, we dropped four catches, there were balls in the outfield we should have cut off.
"It's not a very good advert for the test team so far in this game."
Tendulkar had anchored India's powerful batting line up to an imposing total.
Resuming on 278 for four, just one run behind New Zealand's first innings, Yuvraj Singh (22) was the only wicket to fall in the morning session when he shouldered arms to a Chris Martin delivery and was bowled.
Tendulkar and captain Mahendra Singh Dhoni combined for a century stand until the afternoon session before Dhoni was caught behind by Brendon McCullum for 47 when an Iain O'Brien bouncer brushed his gloves.
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