Pujara's unbeaten 119 took the hosts to 307 for five at the close after winning the toss. Captain MS Dhoni (29) was the other not-out batsman.
Batting at the No 3 spot vacated by the recently-retired Dravid, the 24-year-old Pujara displayed a cool temperament akin to his predecessor as he came in for his fourth Test match and his first since January 2011.
"I wanted to score a hundred on my debut but somehow I missed out so I am really satisfied I've now got it," said Pujara who made 72 in his first Test appearance against Australia in 2010.
"I don't think I can replace Rahul Dravid because he is such a legend. I just want to play my natural game," he told reporters. "I never felt any kind of pressure as I had my preparation and I knew what was coming. I think it's my preparation that has helped me score these runs."
Pujara hit 15 fours and a six during his 226-ball stay and added 125 for the fourth wicket with Virat Kohli (58) to help India overcome the loss of Gautam Gambhir (22), Virender Sehwag (47) and Sachin Tendulkar (19).
"If I can score a double hundred that would be great, but the initial target would be for the team to score 400," said the centurion.
Pujara, who survived a close caught-behind call towards the end of the day, also shared an unbroken 47-run stand for the sixth wicket with Dhoni after the team lost Kohli and Suresh Raina (three) in quick succession.
India, beaten in eight consecutive away Tests in England and Australia last year, lost Gambhir and Sehwag in the morning session and batting great Tendulkar after lunch.
Tendulkar hit two boundaries before he was clean bowled by left-armer Trent Boult. The seamer got the ball to straighten, breaching Tendulkar's defence and taking out his middle stump.
Left-hander Gambhir was the first to depart when he edged Boult to wicketkeeper Kruger van Wyk when trying to run the ball down to third man.
Gambhir put on 49 for the opening wicket with Sehwag who hit nine boundaries and survived two chances during a charmed 41-ball stay at the crease.
Daniel Flynn put down a difficult catch at square leg after the right-hander mistimed a pull off Chris Martin while an edge off Doug Bracewell flew between Van Wyk and captain Ross Taylor at first slip.
Bracewell finally got rid of the dangerous Sehwag when his attempted cut was snapped up by Martin Guptill at second slip.
New Zealand went on to spill more chances later in the day. Pujara was dropped on 60 by Flynn off-spinner Jeetan Patel at short leg while Kohli's edge off the same bowler went between the keeper and Taylor at first slip. Patel was again unlucky when Dhoni survived a difficult chance at slip off his bowling.
"The plan was to put pressure on the younger players and we had a big opportunity to cash in but we missed the bus," said the 23-year-old Boult.
"The first hour tomorrow will be a big opportunity to go at the batsmen and get a couple of wickets."
The second match in the two-Test series is in Bangalore starting on August 31.
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