Three minutes in the second spell, White cut a ball back into the box which Sonkeng turned into her own net and in the 65th minute a White cross from the right was buried with a well timed header from Gregorius to give the Kiwis some breathing room.
Cameroon's Gabrielle Onguene netted from a set piece 10 minutes later to give the Africans scent of a comeback, but that hope was extinguished as New Zealand reasserted control to close out the match and book a quarterfinal against defending Olympic champions USA in Newcastle on Friday.
"We are really happy with the result and the performance as well," said New Zealand coach Tony readings after the match.
"What's most satisfying about reaching the quarterfinals is that we fully deserved it over the course of the three games.
"You work so hard during these tournaments but you don't often get what you deserve so when you do it's really satisfying feeling.
"But for us now, we've reached the quarterfinals and our attention switches straight to Friday and the USA. That's a tough game but it's a game we are looking to win. The prep starts tomorrow and it's full steam ahead."
While the progression to the knockout stages was historic, Readings reiterated that his team were not going to be happy with settling for breaking new ground.
"We came here to play six games - at an Olympics you should be aiming to do that and win the gold medal. To do that we've always said we'll need to grow with each game.
Normally in these tournaments the team that wins it is not necessarily team that starts with a bang they are the ones who grow slowly through the tournament.
"We feel we've done that. We were good against Great Britain and were very unlucky to come away with nothing and were better against Brazil. Today in patches, particularly in the middle section of the game, we played some of the best football we've played for a long while.
"We think we've taken a step up every game and we'll need to do that in the quarterfinal."
The other quarterfinals of Friday see Group E winners Great Britain face Canada, Sweden take on France and Brazil meet World Champions Japan. New Zealand's quarterfinal kicks off at 1.30am on Saturday (NZT).
New Zealand's men also have the chance of reaching the quarterfinals when they meet Brazil in Newcastle at 1.30am tonight.
- HERALD ONLINE