A hat-trick to Rosie White saw the New Zealanders finish third in group A behind Denmark and Canada.
Those two nations drew 0-0 in Hamilton, but still advanced to the quarterfinals.
Colombia had to beat the New Zealanders to have any chance of qualifying for the last eight, but were instead surprised by the home team's tenacity in driving wind and rain which lashed Wellington.
White was the individual star, giving her side a 1-0 halftime advantage after beating Colombian goalkeeper Stefany Castano at the near post in the 44th minute when she latched on to a through ball by Annalie Longo.
Both teams battled in the trying conditions, but it was White who proved the difference.
She scored her second goal in the 81st minute when she intercepted a poor clearance before using her left foot to beat Castano.
Colombia responded within 40 seconds when a long ball placed the New Zealand defenders under pressure, goalkeeper Sarah McLaughlin slipped over and Tatiana Ariza took advantage of the confusion to score.
Five minutes later White settled the issue with a fine header which gave Castano no chance.
In other matches, North Korea drew 1-1 with Germany, a result good enough to send both sides through to the quarterfinals.
Germany were unbeaten in their three matches to top group B, buoyed by a third-minute goal by Alexandra Popp in Hamilton.
North Korea responded in the 58th minute via Jon Myong Hwa, who struck an accurate low, right-foot shot into the far corner.
The other group B match saw Ghana gain their first victory of the tournament, a 1-0 result over Costa Rica in Wellington.
The only goalscoring action came in the 19th minute when Candice Osei-Agyemang broke free down the right before crossing to Deborah Afriyie, who applied an excellent left-footed finish by the far post.
