Graeme Smith's home team restricted the Kiwis to 153-8 after giving them first strike at the Kingsmead and then knocked off the modest target with five balls to spare.
It was New Zealand's only defeat in three matches of the Super Eights and left them level with South Africa on two wins each. Both teams will advance to the semi-finals from group E if India lose to England here later on Wednesday.
If India win and also beat South Africa in their last match on Thursday, a three-way tie will be decided on net run-rates.
England, who lost their first two matches, have been knocked out of the semi-final race.
Pakistan are the first team to make the last four in the 12-nation tournament after beating Sri Lanka and Australia in group F.
The second spot from the group will go to the winner of Thursday's crucial tie between Australia and Sri Lanka in Cape Town.
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Kemp hammered six boundaries and as many sixes in his match-winning knock, ending the match by lofting Mark Gillespie