Dean Elgar failed to join a select group of South Africa batsmen, while New Zealand took three late wickets to leave the first Test at University Oval delicately poised at the end of the fourth day on Saturday.
South Africa were 224 for six at the close of play in a chilly Dunedin, with a lead of 191 runs entering the final day on a pitch that is turning but offering little assistance for the pace bowlers, while proving difficult to score from.
Captain Faf du Plessis was on 56 at the close of play, with Vernon Philander on one after New Zealand had taken three wickets after tea, including Elgar, who was dropped twice and overturned a caught-behind decision during his innings.
The 29-year-old had been seeking to become the sixth South Africa batsman to score a century in both innings of a Test, but fell short when he was caught by Kane Williamson for 89 from offspinner Jeetan Patel. He scored 140 in the first innings.
Temba Bavuma was then bowled by Mitchell Santner for six and Quinton de Kock also bowled by Patel for four in gloomy conditions, with players leaving the field with nine overs remaining.
South Africa had begun the day with a five-run lead and nine wickets in hand and with rain forecast for the final day on Sunday, the hosts needed to make the most of their opportunities to try to dismiss the visitors cheaply.
New Zealand, however, only had themselves to blame for not being in a better position having dropped Elgar twice and JP Duminy once.
The hosts were also further hampered with opening bowler Trent Boult trudging off the field midway through the middle session with a groin strain.
It is also debatable whether batsman Ross Taylor would bat in the second innings after he suffered a torn calf muscle in the first.
South Africa lost just two wickets in the first two sessions when Duminy was trapped in front by Neil Wagner for 39 after lunch, while Hashim Amla was dismissed for 24 early in a disrupted first session.