South African captain Brand was bowled for three runs on an inswinger from his New Zealand counterpart Tim Southee.
Moore was out for zero after covering the ball from Matt Henry.
A tenacious 63-run stand for the third wicket was broken by Jamieson, who trapped Renard van Tonder at first slip on 31, before gently blasting Zubair Hamza’s pull shot to mid-on on 36.
Spinner Mitchell Santner took the turn to wrap up the tail and finished the match with 3-59, equaling Jameson’s six scalps for the match.
Rachin Ravindra, who scored 240 runs in New Zealand’s first innings, was chosen man of the match.
Kane Williamson also shone with the bat and continued his excellent Test form, scoring a century in each innings.