Captain AB de Villiers and opener Quinton de Kock fired the biggest shots for South Africa as they reached 271 for eight in the third one-day international in Wellington.
De Villiers posted a 51st ODI half century in reaching 85 off 80 balls, giving his team a reasonable shot at going 2-1 up in what has been a closely-fought five-match series.
The prolific skipper firstly steadied an innings that was losing direction before opening up over the closing stages with strokes all around the ground.
De Villiers put on 100 for the seventh wicket with allrounder Wayne Parnell (35 off 32) in less than 14 overs, striking seven fours and a six in a milestone knock.
The peerless De Villiers became the 18th player to pass 9000 ODI runs. He boasts both a better average and strike rate than any of his contemporaries.
His 9080 runs have come, remarkably, off 9080 balls while his average is an enviable 54.04.
Like De Villiers, wicketkeeper/batsman De Kock continued an excellent series, passing 50 for the fifth successive ODI innings.
However, he was again guilty of an unnecessary dismissal, caught at deep square leg after reaching 68 off 70 balls. It was repeat of the loose shots that undid innings 69 and 57 in this series.
De Kock and Faf du Plessis (36) both fell in a key over, the 23rd, bowled by New Zealand's best bowler, seamer Colin de Grandhomme (2-40).
Consolidation followed, with De Villiers realising a measured knock was needed in an at-times variable conditions, which weren't entirely comfortable for the batsman.
Opener Hashim Amla (7 off 19) and JP Duminy (16 off 34) were two specialist batsmen who struggled to adapt.
Mitchell Santner (1-45) and Trent Boult (1-47) were also effective for the hosts but not so speedster Lockie Ferguson (1-71), who replaced legspinner Ish Sodhi in the only change from the side that won game two in Hamilton by six runs to square the series.
South Africa also made one change, with paceman Kagiso Rabada usurping allrounder Chris Morris.