Serena, a double Olympic gold medal winner in London, made a series of unforced errors and showed little movement as she laboured her way through a 70-minute match in humid conditions.
"I was out there and I did the best I could today. I was happy to get through that match," said Serena, who won six consecutive games to finish off the match.
"It was just energy. I have been playing a lot, been travelling a lot, been training a lot. It's been a really big summer. Singles and doubles is a tough schedule."
Since the start of Wimbledon in June, she has played 30 matches, with 19 in singles play and 11 in doubles.
But she expects to be in better shape for her quarter-final match-up against Wimbledon semi-finalist Angelique Kerber, the German having progressed with a 6-4 7-6(5) win over Andrea Hlavackova.
"I think it's just a one-day thing. I will get some rest," said Serena, who retained the Olympic tennis women's doubles gold with her older sister Venus in London.
The 14-times Grand Slam winner did spare a thought for her vanquished opponent Radwanska, who is the younger sister of Polish world number three Agnieszka Radwanska.
"They're both great players, great sisters. I absolutely love seeing another pair of sisters in the locker room that's doing so well," she said.
"I think Urszula has worked really hard, and me being a younger sister and Urszula being a younger sister, I'm always looking for her to do well and hoping she does really well."
Australian Stosur dealt with Russian Ekaterina Makarova 6-1 6-3 while top seed Agnieszka Radwanska needed three sets to deal with determined American Sloane Stephens 6-1 4-6 6-4.
The Pole will meet either Li Na or Johanna Larsson, who were made to wait by an evening downpour, for a place in the last four.
Fourth seed Petra Kvitova eased through with a 6-2 6-2 win over China's Peng Shuai and the Czech will now meet wither Caroline Wozniacki or Anastasia Pavlyuchenkova, whose match was abandoned due to rain at the end of the first set, won by the Russian 6-4.