Amla was dropped on 48 but was otherwise typically unruffled as he progressed to 106 not out by stumps, having added 102 with Faf du Plessis to take the Proteas to 325 for four.
Du Plessis (69 not out), Graeme Smith (54) and AB de Villiers (51) also contributed well, while Doug Bracewell (two for 70) was the most threatening bowler.
Alviro Petersen inside-edged the first ball of the match, by Trent Boult, for a lucky four before Bracewell struck Smith a nasty blow on the head in the fourth over. Petersen made 22 before being dismissed by Bracewell, top-edging a weak hook to Jeetan Patel at fine-leg.
Amla got away with a leading edge to his first delivery but Smith struck boundaries in successive Neil Wagner overs. A swept boundary by Amla brought up the Proteas' half-century and though Smith was relieved to see a collection of edges drop safe, he also found the boundary several times before reaching his half-century in the second over of the afternoon.
However, after adding just one more boundary he gloved a ball from Wagner down the leg side and was caught by wicketkeeper BJ Watling.
Jacques Kallis scored his only runs with back-to-back boundaries off Bracewell before edging the same bowler to Watling, but Amla reached a 92-ball half-century just two balls after being badly put down by Kane Williamson at gully. AB de Villiers made a fluent start and though he faced a confident lbw shout from Jeetan Patel, he and Amla added 54 by tea.
Martin Guptill missed a chance to run out De Villiers two balls after the restart, allowing the wicketkeeper to press on towards his half-century. He got there by taking seven off an over from Wagner, including his sixth four, but swatted Patel straight to Williamson at mid-wicket in the next over to end a stand of 86.
Du Plessis edged just short of slip via Watling's pads but was quickly into his stride with the help of four overthrows from Boult and a boundary off Bracewell. Amla also escaped a run-out threat as he moved into the 90s, and was happy to accumulate singles as Du Plessis pushed the scoring along.
The latter was beaten all ends up by the last delivery before the new ball and then escaped when, on 42, Hot Spot showed he gloved Boult to Watling but New Zealand did not refer the decision. Amla reached a 187-ball ton with his eighth four and Du Plessis brought up his 50 and the team's 300 in even greater style with a huge six off Bracewell.
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