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The 29-year-old Amla was unbeaten on 106 at the close of play, his 19th test century, and was once again the rock of the innings as he shared in three big partnerships after the home team won the toss and elected to bat.
The right-hander, who went to the crease with the score 29 for one, had a lucky escape on 48 when he was dropped in the gully by Kane Williamson off left-arm paceman Trent Boult.
The elegant Amla began slowly, bringing up his half-century off 92 balls, and only hit eight fours all day.
He was aided in his efforts by Graeme Smith, AB de Villiers and Faf Du Plessis who all made half-centuries on a slowish wicket which offered increasing spin and variable bounce as the day wore on.
South Africa, 1-0 up in the series, lost opener Alviro Petersen for 21 before Smith and Amla combined for a second-wicket partnership of 92.
Captain Smith was then caught down the leg side by keeper Bradley-John Watling off left-arm seamer Neil Wagner.
Smith struck 54 off 88 balls with eight fours and New Zealand's spirits were further lifted when Jacques Kallis departed for eight soon after.
De Villiers and Amla steadied the innings with a fourth-wicket stand of 86 with the former reaching 6,000 test runs. He made 51 before he slapped a delivery from off-spinner Jeetan Patel to Williamson at mid-wicket.
"He (Amla) is a wonderful player," De Villiers told reporters. "I love batting with him, he is a really calm man.
"He is the rock at number three for us. The whole lineup fed off him today and it got us into a strong position."
The dismissal of De Villiers left South Africa on 223 for four and Amla and Du Plessis then combined for an unbroken fifth-wicket partnership of 102.
Du Plessis was in a particularly positive frame of mind as he thumped 69 not out with eight fours and a six.
Paceman Doug Bracewell was the pick of New Zealand's attack with two for 70.
"Hashim batted really well and it shows that against a good side you have to create more opportunities by stringing a lot more good balls together for a period of time, said Wagner.
"The key word today for us was execution and it wasn't always there. You are never out of it though - we have to fight back tomorrow and be ruthless."