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Poland's Radwanska, who reached last year's Wimbledon final, battled fearsome temperatures topping 40 degrees Celsius at the Olympic Park complex to brush past Japanese veteran Kimiko Date-Krumm 6-4, 6-3.
While Radwanska reached the last eight, the major talking point was not the standard of play but the blistering heat ahead of next week's first Grand Slam event of the year at the Australian Open in Melbourne.
"I think this is too hot to play tennis," said Radwanska. "Even for players, for ball kids, for the people sitting out there, I think it's just too hot.
"I think I would prefer to go on court at 23:00 and play a match."
The temperature climbed to 41.4 Celsius in the early afternoon at the tournament, with readings expected to rise further.
Radwanska said she was trying not to alter her game in the conditions but admitted she had more of a battle with the heat than her opponent.
"I think the first main thing is just to stay calm and trying to survive," she said.
Li Na, who beat Japanese qualifier Ayumi Morita 6-1, 6-0, said she was reaching for the ice towels at each changeover.
"I was feeling like playing in a sauna. At the first changeover I didn't use an ice towel. I was feeling the heat was coming and I was like, 'What's going on'?" Li said.
"At the next changeover I was like, 'OK, I need an ice towel every changeover, otherwise I don't know how to play on the court'.
"It's too hot, but I'm still in the tournament, so it's good news."
The tournament has an extreme heat policy but it only comes into effect at the discretion of the tournament referee.
Tennis officials apply a complex formula factoring in heat, humidity, and wind to determine when on-court conditions become too stressful for the players.
