Moreki was one of six uncapped players, including captain Neil Brand, named in the Proteas line-up after South Africa's leading players stayed at home to take part in a domestic Twenty20 competition.
Brand won the toss and chose to bowl, and Moreki rewarded his skipper's decision by trapping opener Devon Conway (one) lbw from the first ball of the second over.
Moreki has taken 192 wickets in 93 matches in South Africa's domestic first-class competition and is playing his first Test at the age of 30.
The right-armer was effective bowling round the wicket to the left-handers and across the right-handers as the ball ducked about a little on a hard, dry pitch at the Bay Oval in Mount Maunganui.
Moreki shared the new ball with Duanne Olivier, one of the most seasoned members of this South Africa team in his 16th Test.
Tom Latham (20) also fell before lunch, caught by wicketkeeper Clyde Fortuin from the bowling of Dane Paterson in his fourth Test.
Paterson bowled just above medium pace but his accurate line and consistent length caused trouble for the New Zealand batters.
Probing with several deliveries on a good length and passing across the left-hander, Paterson's approach bore fruit when Latham pushed forward to a ball moving away outside off and presented Fortuin with his first catch in Test cricket.
Williamson returned to the New Zealand line-up on Sunday after recovering from a hamstring injury and successfully negotiated a testing first morning, going to the break unbeaten on 26.
Rachin Ravindra was 16 not out as New Zealand made slow progress in the 25 overs before lunch.