Black Caps openers Finn Allen and Tim Seifert had come out swinging after their side batted first.
Allen hit four fours and two sixes for his 17-ball 31 before Marco Jansen (4-40) crucially took three wickets in quick succession to bring South Africa back into the game.
The seamer sent back Seifert for 13 in his first over before dismissing Rachin Ravindra and Allen in the space of four balls in the final over of the powerplay.
Mark Chapman and Daryl Mitchell rebuilt the innings and New Zealand were scoring at around 10-an-over for more than half the innings.
They were 138-5 when Chapman fell to the last ball of the 14th over, having made 48 from 26 balls, with their opponents able to drag it back from there and restrict New Zealand to 175-7.
It appeared short at the halfway stage and within a few balls of the reply, that was confirmed.
Markram and De Kock got after opening bowlers Matt Henry and Lockie Ferguson from the off, then gave Jacob Duffy the same treatment when he came on.
Ferguson bowled De Kock with a slower ball but there was no slowing the Proteas as Markram continued his charge, first alongside Rickelton then Dewald Brevis, who both made 21.
Markram was able to watch from the non-striker’s end as David Miller finished the job in style – launching Ferguson into the stands for a monstrous six.
South Africa will official qualify for the Super 8s if the United Arab Emirates fail to beat Afghanistan on Monday.