Cook Islands winger Esom Ioka scored three tries, putting his team ahead 16-12 after New Zealand opened with two quick scores.
The Kiwis recovered to win 82-16 in Christchurch on Sunday, earning their first Rugby League World Cup victory after a two-point loss to Australia in the opener.
New Zealand had also lost fullback Kalyn Ponga in that defeat after he broke his arm.
Jamayne Isaako, the NRL's top point scorer in 2026, scored a hat-trick, kicked 13 of 14 conversions and finished with 30 points.
Christchurch debutant Tanner Stowers-Smith scored twice in front of a sold-out crowd.
"We got punched in the face," Kiwi coach Stacey Jones said of Ioka's hat-trick.
Cook Islands coach Karmichael Hunt hoped the tournament could help talented youngsters from his team find NRL opportunities.
New Zealand will face Fiji on the Gold Coast on Saturday, while the Cook Islands meet Australia in Townsville to complete pool play.