Du Plessis made 58 off 40 balls as South Africa made 179 for six after being sent in on a cool evening.
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Finch hit 114 and Smith 105 in Australia's powerful 374 for six and restricted the tourists to 308 for eight.
The Black Caps finished on 179-5 after being set a target of 176 in a match where the innings were reduced to 16 overs because of persistent showers.