Despite losing its first two games of the group stage, New Zealand finished third in Group E and qualified ahead of North Korea on goal difference.
The two best third-placed teams advanced, along with the top two in each of the three groups.
Rebecca Smith opened the scoring for New Zealand in the 43rd minute, an own goal from Ysis Sonkeng made it 2-0 in the 49th and Sarah Gregorious added a third for New Zealand in the 62nd.
Gabrielle Onguene got a consolation goal for Cameroon in the 75th minute at the City of Coventry Stadium.
The result left Cameroon bottom of the group with no points, having lost all of their games.
Tony Reading's team now faces the defending champion United States in Newcastle on Friday.
Cameroon had the first chance of the game in the third minute when Onguenne unleashed a fierce long-range shot, but it flew wide of the goal.
New Zealand soon began to stamp its authority on the match and Abby Erceg was unlucky not to score in the 17th minute when a cross from Ria Percival zipped across the face of the goal.
Smith finally put New Zealand in front two minutes before the break, heading past Cameroon goalkeeper Anette Ngo Ndom to score her team's first goal of the tournament.
The second goal resulted from a string of Cameroon defensive errors. Ndom came out of her area to try and stop Rosie White who was storming toward the goal. White skipped past the goalkeeper and her cross was deflected into the net by Sonkeng.
Gregorius made it 3-0 with a 62nd-minute header.
Onguenne pulled one back for Cameroon with a header but it was too little too late for the Africans.
The New Zealand men's team has the chance to make it a historic quarterfinal double for New Zealand's Olympic football teams when it faces Brazil on Wednesday, needing a win to make it through to the knock out stages.