Man of the match Steyn took 3-48 and Morkel 2-36 as New Zealand were bowled out for 211.
GALLERY: South Africa v New Zealand, second Test
They needed only a combined 34 deliveries to take the last five wickets. Steyn took 5-17 in the first innings and had match figures of 8-65.
Scorecard: South Africa v New Zealand, second Test, Port Elizabeth
Video highlights: SA v NZ, second Test, day 4
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 10th 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.
Teams
South Africa:
Graeme Smith (captain), Alviro Petersen, Hashim Amla, Jacques Kallis, AB de Villiers (wk), Francois du Plessis, Dean Elgar, Robin Peterson, Rory Kleinveldt, Dale Steyn, Morne Morkel
New
Zealand: