Jamayne Isaako scored 38 points against the Cook Islands, setting a record for the most points by a New Zealander in a Test match.
His hat-trick helped New Zealand recover from its opening-round loss to Australia with an 82-16 victory.
The Kiwis face Fiji in Round 3, with the winner keeping its World Cup campaign alive.
Australia secured a semi-final place by beating Fiji 36-22 in Brisbane.
Fiji's Kurt Donoghoe earned Player of the Match honours, recording three try assists, three line-break assists and a tackle break.
Dolphins winger Jack Bostock and the Kangaroos also progressed to the knockout stages.
England revived its semi-final hopes with an 82-0 win over France, led by a hat-trick from Dolphins centre Herbie Farnworth.
Farnworth ran 168 metres, including 40 post-contact metres, and made nine tackle breaks.
England must beat Papua New Guinea in Round 3, while its semi-final hopes also depend on the Tonga-Samoa result.
Tonga defeated Lebanon 36-0 to extend its unbeaten run under Dolphins head coach Kristian Woolf.
The win sets up a Round 3 clash with Samoa at CommBank Stadium, with both sides seeking a semi-final place.
Samoa beat Papua New Guinea 64-6, as Dolphins fullback Hamiso Tabuai-Fidow scored twice and made 154 metres, seven tackle breaks and two line breaks.
Round 3 will determine the remaining semi-final places.