The 21-year-old German, who had never been past the second round of a grand slam before this year's tournament, had Radwanska on the ropes when she led 5-2 in the deciding set before injury cruelly struck at 5-4.
The 82nd-ranked Friedsam, who beat U.S. Open finalist Roberta Vinci in the third round, then took an injury time out and had strapping applied to her left leg.
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"I felt pain early in the third set, but I didn't know whether it was cramping or an injury," Friedsam told reporters. "It's a bad feeling."
Radwanska held serve in the next game, but Friedsam appeared to exacerbate her injury.
The German was given a time warning on the first point of the next game when she was unable to serve and broke into tears. She managed to play on and was 15-40 down when she appeared to strain her right hamstring.
"The big thing was the serve. I couldn't push off the leg," she added.
"It was 5-5 when I really felt it, and I had never had it so I didn't know how to handle it."
Chair umpire Felix Torralba then sanctioned her again for exceeding the time limit and awarded Radwanska a point that sealed the game and gave her a 6-5 advantage.
The trainer again visited Friedman, who was in constant tears on her chair, but she managed to hobble out and complete the match and was given a massive cheer by the Hisense Arena crowd for her efforts.
Radwanska will now meet Spain's Carla Suarez-Navarro, who beat Australia's Daria Gavrilova 0-6 6-3 6-2, in the last eight.
The Pole, who had to play through the pain of a leg injury in the third round, said she had felt a twinge of sympathy for her hobbled opponent but also needed to be ruthless to advance.
"Of course, it's hard to see that," said Radwanska, a semi-finalist at Melbourne Park in 2014.
"It's tough because you know someone is struggling. But you really have to play your game because someone is still fighting.
"I had no choice, just to fight till the end for each point."
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