Man of the match Dale Steyn took three for 48 and Morne two for 36 as New Zealand were bowled out for 211.
They needed only a combined 34 deliveries to take the last five wickets. Steyn took five for 17 in the first innings and had match figures of eight for 65.
New Zealand were bowled out half an hour before lunch on the fourth day as South Africa completed a 2-0 series victory. They won the first Test in Cape Town by an innings and 27 runs.
Dean Brownlie and BJ Watling put on 98 for the fifth wicket before Brownlie was caught behind off Jacques Kallis for 53 in the tenth over of the day.
The new ball was taken three overs later and the rest of the batsmen folded quickly.
Colin Munro was caught at third slip off Morkel in the first over with the new ball and Watling's defiant innings ended when he was bowled by Steyn for 63 in the next over.
Watling and Brownlie were among the few players to shine for New Zealand in a one-sided series.
Brownlie made a century in the second innings in the first innings, while Watling made 63 in both innings in Port Elizabeth.
The win consolidated South Africa's position at the top of the world rankings.