Martic defeated Kurumi Nara 6-2, 3-6, 6-0 in earning a spot in the quarterfinals. She had just one break point in the first and third sets and finished the match by taking 68 percent of the points in the final set.
The schedule called for five matches, including a first-rounder pitting top-seeded Agnieszka Radwanska and Akgul Amanmuradova.
That match was to have been played Tuesday but rain and lengthy earlier matches led organizers to put it in the middle of Wednesday's schedule.
Radwanska and Amanmuradova did get started, with Radwanska taking a 6-2, 3-3 lead, but rains hit again and play was stopped for the day.
They will be first up Thursday with the winner taking on Karolina Pliskova later in the day.
Because of rain delays, the match time was nearly 4 1/2 hours.
In the other second-round match completed Wednesday, qualifier Hsieh Su-Wei beat Casey Dellacqua 6-1, 7-5 behind six service breaks.