South Africa captain Faf du Plessis, who won the toss and chose to bat after overnight rain delayed the start of play by 30 minutes, was on 33 and Temba Bavuma on 13 when the decision to end play was made.
The hosts had dismissed Hashim Amla for 50 after lunch just before the rain returned and forced the players off the field at about 3.20 p.m. (0220 GMT) midway through the middle session.
While the rain had stopped, umpires Rod Tucker and Bruce Oxenford abandoned play shortly after 5.45 p.m. after they inspected the wicket and outfield.
New Zealand had reduced South Africa to 71 for three at lunch, which was taken five minutes early when a rain shower swept across the ground.
Matt Henry, who was brought back for the injured Tim Southee, dismissed debutant opener Theunis du Bruyn for a duck and had JP Duminy caught in the deep about 20 minutes before lunch.
Colin de Grandhomme had the first session's other wicket when he bowled Dean Elgar for five as the hosts, who were also without Trent Boult, reduced the Proteas to 5-2 in the fourth over.
Amla was the only other wicket to fall when he played around a de Grandhomme yorker shortly after he had reached his 32nd test half century.
Rain is forecast for the Hamilton area throughout the test, which New Zealand must win to level the series after South Africa won the second game in Wellington by eight wickets.
The first test in Dunedin ended in a draw with the entire final day washed out.
(Reporting by Greg Stutchbury; Editing by Peter Rutherford)
