Play was suspended for more than six hours before the match was postponed until Saturday.
Wozniak is attempting to become the first Canadian to reach the quarterfinals in 20 years. Patricia Hy-Boulais was the last to do it, losing to Monica Seles in the quarterfinals in 1992.
Earlier, top-seeded Victoria Azarenka withdrew because of a left knee injury.
Azarenka, who had a first-round bye, was tied at 3-3 in the first set when her second-round match against Austria's Tamira Paszek was suspended by rain Thursday night.
"I hope it is a short-term thing, but I don't know," Azarenka said. "I have to kind of stop and just take a little bit of time off because I absolutely had no rest for a long period of time, and just to settle it down and to go from there. But hopefully there is nothing serious."
The 23-year-old Belarusian, ranked No. 1 in the world, won two Olympic medals in London, taking gold in mixed doubles and bronze in singles.
"The last week took so much energy out of me that I absolutely had no time to recover and my body is feeling it," Azarenka said. "It was a little bit too much of a transition for me to come and start the training so I just felt my knee was not in good shape."
She said the knee didn't bother her during the Olympic tournament, played on grass at Wimbledon.
"It actually just happened when I came here," Azarenka said. "When I started to train, the first time I hit it was a little bit of a compensation from the other court and it just happened."
Azarenka confirmed that she would not play next week in Cincinnati. She said getting herself ready for the U.S. Open is her top priority.
"The first is to get healthy, to make sure that I'm a hundred percent ready to play at the (U.S.) Open," Azarenka said.
Russia's Maria Sharapova, ranked No. 2 in the world rankings, withdrew earlier from the tournament because of a virus. The Olympic silver medallist and French Open champion was the third seed.
"Her performance today was extraordinary," Bartoli said. "When she plays like that she makes it very hard for her opponent to compete. I had break opportunities, and every time she served an ace."