"We've got a mountain to climb," assistant coach Adrian Birrell said after South Africa were reduced to 80 for five at stumps on day four, 95 runs short of making New Zealand bat again.
"Kane Williamson's innings was a fabulous innings and you've got to sometimes doff your cap to a good player."
With Colin de Grandhomme also firing with the bat with a quickfire 57, New Zealand started the day seven runs ahead and added 168 before they were all out for 489.
De Grandhomme then ignited the assault on the South African top order in the sixth over when he removed Dean Elgar for five and later took a smart catch to dismiss Hashim Amla for 19.
Theinus de Bruyn was run out for 12 in a bizarre mid-pitch collision when he was ball-watching and veered into Amla, leaving the batsmen sprawled on the ground.
Birrell said the Proteas' batting mayhem was part of the flow-on effect from Williamson's marathon innings.
"One hundred and sixty two overs is a long time to be in the field and it was a hard day. Players are fatigued and to bat on the back of that is always going to be difficult." he said.
"Once they got ahead of us by 100 runs it was always going to be a survival match."
Faf du Plessis and Quinton de Kock were both 15 not out at stumps. Birrell said they were looking for de Kock, who set up their eight wicket win in the second Test to go 1-0 up in the series, to go on the attack when play resumes.
"If he goes into a defensive mode it's probably the worst thing for him, so we look for him to score. Ninety-five runs, we start getting the positive, and every run in the positive is one they have to get and is time wasted."
The Zimbabwe-born de Grandhomme was not required for the first Test when Elgar scored 140 and 89 but in four innings since then he has removed the opener for scores of nine, 17, five and five.
The loss of Elgar should have signalled to South Africa it was time to dig in but instead wickets fell at regular intervals.
After the de Bruyn run out for 12, Amla progressed from eight to 19 when he edged a ball from Jeetan Patel which rebounded off Watling's glove to the sole slip de Grandhomme.
Patel also bowled JP Duminy (13) and Matt Henry had Temba Bavuma caught behind for one to leave South Africa 59 for five, before du Plessis and de Kock survived 12 overs to stumps.
Should New Zealand level the series it will be a remarkable turnaround from the eight-wicket, three-day hiding they suffered in the second Test.
It would also have been achieved without injured strike bowlers Trent Boult and Tim Southee as well as senior batsman Ross Taylor.
After Williamson and Mitchell Santner (41) consolidated with a patient 88-run stand for the fifth wicket, de Grandhomme finished the innings with a flourish, smacking five fours and two sixes in his 70-ball stay.
Williamson had looked assured throughout until he hooked Morne Morkel to Vernon Philander at long leg after 449 minutes in the middle.
He faced 285 deliveries and struck 16 fours and three sixes.
BJ Watling contributed 24, Henry 12 and Patel five before the innings folded with de Grandhomme gloving a Morkel flyer through to wicketkeeper de Kock.
Morkel took four for 100 and Kagiso Rabada four for 122 while de Kock held five catches behind the stumps.