Niukore scored a double in Sunday's big win at CommBank Stadium, with the Pacific nation proving too skilful for the Rhinos.
Sporting a side with no NRL players and only a few State Cup members, South Africa led 6-4 early and did well to stay in the contest.
But when prop Davvy Moale crashed through three defenders to make it 10-6, South Africa's hopes of returning to the World Cup for the first time since 2000 quickly evaporated.
Esan Marsters and Niukore both scored within five minutes of each other just before halftime.
And in the second half, it was a demolition job.
Five-eighth Cassius Cowley scored two tries and Niukore finished off his double when he ran another great line to go in.
Brendan Piakura produced the try of the game when he split two defenders on halfway and stepped South Africa's fullback to score.
Cronulla star KL Iro also rounded out a 60-metre effort when Rua Ngatikaura threw an off load and Cassius Cowley went into space.
And with that, Cook Islands were celebrating their fourth World Cup appearance.
Australia and New Zealand will kick off the five-week tournament in Sydney on October 15, before it culminates in a final at Suncorp Stadium in mid-November.
The 10-team competition will split sides between three groups in the preliminary stage.
Australia and New Zealand will be in a four-team Group A, with the top two to qualify for the semi-finals.
The teams across Groups B and C will play against the three sides in the opposing group, and effectively form one six-team ladder with the top two to progress.