Steyn took three wickets with his first 11 balls of the innings and finished with five for 33 as New Zealand were bowled out for 195 despite a defiant innings of 76 by Henry Nicholls.
It looked as though it would be easy for the hosts when Steyn and Vernon Philander ripped through New Zealand's top four batsmen for seven runs inside the first four overs of the innings.
But the left-handed Nicholls, playing in his sixth Test match, displayed courage and concentration in making his second Test half-century and highest score to delay South Africa's celebrations.
Nicholls was last man out after batting for 255 minutes. He faced 140 balls and hit 12 fours.
Steyn bowled Tom Latham with the first ball of the final innings, then had Latham's fellow opener, Martin Guptill, caught at first slip, also for a first ball duck, off the final ball of the over.
The magnitude of New Zealand's task was shown in the second over when captain and first innings top-scorer Kane Williamson was struck a painful blow on the left hand by a ball from Vernon Philander which reared up off a good length. He required lengthy treatment before resuming his innings.
In between, Ross Taylor was trapped leg before wicket by an unplayable ball from Steyn, which skidded through barely above ankle height off a pitch with increasingly unpredictable bounce.
Latham tried to leave the ball from Steyn that bowled him but it lifted sharply and bounced off his raised bat onto the stumps.
Guptill, a heavy scorer in limited overs international cricket, again raised doubts about his technique in Test cricket when he edged a ball that seamed away from him.
Watling was dropped by Stiaan van Zyl at third slip off Philander when he had five and two runs later seemed lucky to get the benefit of the doubt from television umpire Richard Illingworth when the South Africans appeared convinced that he had gloved Philander to De Kock.
South Africa added 27 runs in 13 overs, for the loss of Philander's wicket, before declaring their second innings on 132 for seven. Temba Bavuma made 40 not out. Tim Southee bowled Philander and finished with three for 46.
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