The Pole, who is seeded second in the August 4-13 tournament, will replace Victoria Azarenka as world number one if she wins the $2 million event and the reigning Australian Open champion fails to reach the quarter-finals.
Radwanska, who was stretched all the way by German Mona Barthel on a day of upsets, came from a break down in the final set to win 4-6 6-3 7-6 (5).
At least Radwanska, who plays Chanelle Scheepers in the third-round, completed her match on a day marred by rain.
Azarenka will be forced to wait until Friday at the earliest to finish her match against Tamira Paszek after it was postponed on Thursday night due to rain.
The second round encounter is tied at 3-all in the opening set.
"I can't remember last time I had a match like this, you know, maybe not since I was a junior," the 24-year-old Serbian, who won this event in 2006, said.
"It's obviously tough, you know, but I just want to try and put it behind me and try to forget about this because that's the only thing you can do."
American Varvara Lepchenko upset 12th seed Dominika Cibulkova 6-4 6-4 in a rain-interrupted match and will play either Caroline Wozniacki or Kiki Bertens on Friday. Wozniacki, the 7th seed, leads 7-5 4-0 in the match still to be completed.
China's Li Na, the 10th seed, ended the hopes of Wozniak's doubles partner and fellow Canadian Eugenie Bouchard 6-4 6-4 just before rain ended play for the day. She said the conditions, which are expected to continue on Friday, made it difficult for all the players.
"It's really tough. I think I was warming up like a hundred times," she said. "You never know when you can come to court, so you always have to ... stand up every second."
15th seeded Sabine Lisicki of Germany needed treatment from the trainer during her 3-6 6-3 6-4 loss to Spaniard Carla Suarez Navarro.
Ninth seed Marion Bartoli, who advanced on Thursday by beating China's Peng Shuai 6-1 6-3, opens play on Friday against former Wimbledon champion Petra Kvitova. Reigning US Open champion Sam Stosur faces 16th seed Lucie Safarova on the second stadium court.