Dunedin: South Africa were 38 for one, leading New Zealand by five runs, when bad light stopped play on day three in the first Test in Dunedin on Friday.
Hashim Amla was not out 23 with Dean Elgar on 12.
Earlier, New Zealand were all out for 341 with Kane Williamson scoring 130, Jeet Ravel 52 and BJ Watling 50. For South Africa, Keshav Maharaj took five for 94.
Half-centuries from Jeet Raval and BJ Watling, propelled the hosts to a 33-run advantage before they were all out for 341 after tea on day three.
The last time New Zealand led the Proteas at halfway was five years ago, also in Dunedin, when they were 35 ahead before the match ended in a draw due to rain.
New Zealand inched towards South Africa's 308 after Williamson was dismissed for 130.
It started a run of three wickets for 27 runs before Neil Wagner levelled the scores with nick off Vernon Philander that brushed Quinton de Kock's fingers on its way to the boundary.
Trent Boult put New Zealand in front with a single off Keshav Maharaj before he was bowled by the spinner to bring the injured Taylor back to the middle for a last stand with Wagner.
With the lead secured, Wagner opened up with a six and two fours off consecutive balls from Philander and added another six and four off Maharaj.
Taylor, crippled by a calf tear that forced him from the field when on eight early in the innings, was barely able to run when he returned to the crease.
After an early single off Maharaj, his one other scoring shot was from clouting Morne Morkel over the ropes for six.
The 27-year-old Maharaj, in his fifth Test, ended the innings when he had Wagner caught by JP Duminy for 32 to finish with five for 94, his first five wicket Test haul.
On a slow track, Raval with 52 and Watling who made 50 were the only other New Zealand batsmen to reach the half-century mark.
Williamson was patience personified in his innings in which he faced 241 deliveries in 280 minutes.
When he resumed on Friday he faced 27 deliveries before adding to his overnight 78.
South Africa - 308 (Elgar 140, Bavuma 64, Boult 4-64) and 38 for 1 (Amla 23*)
New Zealand - 341 (Williamson 130, Raval 52, Watling 50, Maharaj 5- 94)
Status - South Africa lead by five runs