Martens delivered a long strike from 25 yards outside the box for the Netherlands' first ever goal at a World Cup. The Netherlands are one of eight teams making their debut at the tournament.
Danielle Van De Donk of the Netherlands appeared to score at the quarter-hour mark but was ruled offside.
Goalkeeper Loes Geurts was able to deflect a free kick from New Zealand's Ria Percival which was directed at the top corner. Martens would find the net shortly after.
Neither 'keeper was seriously tested in the second half.
Earlier in the day, Canada's Christine Sinclair converted a stoppage-time penalty for a 1-0 win over China.