Jean-Paul Duminy, who hit an unbeaten 43-ball 86 with ten boundaries and three sixes, helped his team to 170-6, while Hashim Amla made 41 at Zahur Ahmed Chowdhury stadium.
New Zealand needed seven off Dale Steyn's last over and three off the final delivery, but Ross Taylor was run out as the Kiwis managed 168-8.
Styen finished with 4-17.
For New Zealand, Taylor scored 62 and Kane Williamson notched a 35-ball 51.
Sri Lanka play the Netherlands in the same group later Monday.
The final team in the group is England.
Title-holders the West Indies, India, Pakistan, Bangladesh and Australia form Group Two.
The top two teams will advance to the semi-finals.
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