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 German, 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.
Radwanska held serve, 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.
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 either Spain's Carla Suarez-Navarro or Australia's Daria Gavrilova in the last eight.