At tea, New Zealand were 139 for five with a lead of 48 runs and only five wickets remaining.
Opener Jeet Raval, battling to extend the New Zealand innings, was on 72 in a 49-run partnership with BJ Watling who was not out 14.
It is Raval's highest Test score, surpassing the 55 he posted twice against Pakistan last November.
When South Africa resumed the day at 349 for nine, Morkel extended their first innings lead to 91 and pushed the score to 359 in a record 10th-wicket stand with Vern Philander.
After removing the out-of-form Tom Latham for six, Morkel captured the key New Zealand wicket of Kane Williamson for one and followed up with Neil Broom for 20. All three were caught behind the stumps.
Williamson's match total of three is his worst performance in any of his 60 Tests where he has batted in both innings.
The 1.96 metre (6ft 5in) Morkel, getting bounce and seam movement, found the faintest of edges from Williamson which eluded umpire Kumar Dharmasena but was detected by the "hotspot" technology.
Keshav Maharaj accounted for first innings centurion Henry Nicholls for seven and followed with the wicket of Jimmy Neesham (four) to have two for 26 off 14 overs.
Neesham had attempted to clout a loose ball to the boundary only to be caught by a leaping Faf du Plessis at mid-wicket.
It left New Zealand at five for 95 when Watling joined Raval to get New Zealand into credit.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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