New Zealand had lost their two reviews by the 29th over, and were powerless to react 13 balls later when the umpires missed a faint edge which should have seen du Plessis caught behind for 16.
At stumps, the Proteas' captain had advanced to 33, Temba Bavuma was on 13 and South Africa were 123 for four.
Resting injured De Kock not an option: Du Plessis
Only 41 overs were possible with nearly three hours play lost because of rain and further showers forecast for the remaining four days have diminished the chance of a result.
South Africa, starting the Test with an unbeatable 1-0 series lead, were two down for five runs in the fourth over after winning the toss and electing to bat.
It would have been three for 28 if not for the first New Zealand judgement error of the day.
They decided not to review a rejected Neil Wagner lbw appeal against JP Duminy on seven when the ball tracker technology signalled a direct hit.
Duminy went on to make 20 in a 59-run stand for the third wicket with Amla before being removed just before lunch.
Amla went soon after the break for 50, his 32nd half-century and was left 67 runs short of becoming the fourth South African to total 8,000 career runs.
After New Zealand's first review blunder they over-compensated by wasting two reviews on correctly-called not out decisions which left them helpless to contest another decision for 51 overs when the second new ball became available.
Kiwis dealt Southee blow, Proteas sweat over de Kock fitness
The magnitude of the error became apparent minutes later when umpire Bruce Oxenford missed a faint edge from du Plessis off the luckless Wagner which was taken by BJ Watling behind the stumps.
It continued a charmed run at the crease for du Plessis who has seven fours in his 33, including from edges over the wicketkeeper, through the slips and just wide of gully.
But there were mixed fortunes for du Plessis at the start of the day.
He won the toss for the eighth consecutive time in New Zealand but then saw both openers fall cheaply.
Debutant Theunis de Bruyn's introduction to Test cricket ended with a third-ball duck when he edged Matt Henry to Tom Latham at second slip.
Dean Elgar was bowled for five when he shouldered arms to a Colin de Grandhomme ball that came back from outside the off stump.
Henry also accounted for Duminy to have figures of two for 25 while de Grandhomme bowled Amla and finished the day with two for 43.
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