They will meet Japan in Sunday's quarterfinals at 10.36 am.
Britain had lost 14-19 to Fiji earlier but recovered with a 36-0 win over Papua New Guinea.
Wing Michaela Blyde put her speed to work to get the Black Ferns out to a 10-0 lead at the break, her second try coming 1.34min into extra time after New Zealand started a move from their line.
Strong defensive work rattled the British side as they got into the New Zealand 22m area, but they could not get over the line. Jazmin Felix-Hotham pulled off another superb defensive tackle when a try looked possible.
From a scrum on their line, the Black Ferns moved the ball around, and space opened up to allow Stacey Waaka-Fluhler to show her speed to lift the score to 15-0.
Then as the game closed out, and England yielded the ball when making a last-ditch effort to score, New Zealand secured, and Shiray Kaka did some outstanding lead-up work to link with Sarah Hirini, who then ensured replacement Portia Woodman had a chance to score to complete the win.
Earlier, the Black Ferns Sevens made light of the invited team Papua New Guinea by scoring a 58-0 win. Michaela Blyde scored twice, Stacey Fluhler three times, Shiray Kaka, Sarah Hirini (twice), Alena Saili and Jorja Miller crossed for tries.
And in their second game, they kept their goal-line intact with a 27-0 win over Fiji. Blyde put in three outstanding defensive tackles to maintain that defensive effort.
Risi Pouri-Lane, Blyde, Jazmin Felix-Hotham, Fluhler and Hirini scored tries.
In the other quarterfinals, Australia will play Britain, the USA will play Fiji, and France will play Ireland.
Catch up on all the action and results HERE.