New Zealand pride themselves on their slick possession based game, but for long periods in the first half their passing lacked accuracy, and their back up plan - a long outlet ball to powerful striker Hannah Wilkinson - was easily dealt with by the Dutch.
The Ferns looked most dangerous from set plays, where they looked to crowd out the Dutch goalkeeper and use their physical advantage, but they were only able to carve out a few half-chances.
Midfield technician Annalie Longo was the Ferns' most industrious player and her influence on the game grew as the match progressed.
The Netherlands, who were appearing in their first World Cup, dominated possession in the first half and sat back and defended their lead in the second.
And despite the Ferns pouring forward in numbers and dominating the second half possession stats, New Zealand never really looked like breaking down the experienced Dutch rearguard.
The one moment of quality that decided the match came in the 33rd minute when Martens found some space on the edge of the area before expertly curling the ball past Ferns goalkeeper Erin Naylor. It was a goal worthy of a World Cup win and the mainly neutral crowd erupted in appreciation.
"We're happy with the way we played today," said coach Tony Readings.
"We played a team ranked 12th in the world and I thought we were the better team. The first half was quite tight and even but in the second half we were by far the better team and sometimes you don't get your rewards but if we keep playing like that throughout the tournament we will be fine.
"We also know we can play better than that. That was the challenge at halftime, and we did, we dominated the second half and were unlucky not to get a few penalties or have a few things bounce our way."
The Ferns now face a difficult path to progress to the knockout stages. Next they face hosts Canada (ranked 8th) on Friday, before China (16th) next Tuesday.
The top two from each group progress to the Round of 16 as well as the four best third-placed finishers and the Ferns will need at least three points to stand any chance of going through.
"We said the games will all be tight in this group and I think today's games have both proved that. We saw Canada play this morning and they had some weaknesses but found a way to win and that's something we need to learn how to do."