The former world number one was broken in the eighth game of the first set and as the injury began to affect Williams, Radwanska tore through the second to clinch a famous victory.
"I was having a big problem with it and there was nothing I could do today about it," Williams said.
"It was at the end of the first set I couldn't hit my backhand any more.
"I was playing pretty good tennis in the first match and thought I could move on from there, but today was really frustrating."
Williams, whose ranking has dropped to 54 after almost three months out, had looked in good form when she beat sixth seed Ana Ivanovic in round one.
But against Radwanska, the American could do nothing and her injured wrist clearly affected her more as the match went on.
World number 95 Radwanska, who reached the fourth round at Wimbledon, broke for 5-3 and then saved two break points to serve out for the first set.
From then on, with Williams in pain, there was only ever one winner and the Pole wrapped up victory in one hour to set up a clash with top seed Elena Dementieva of Russia.
Second seed Nadia Petrova was beaten 6-4 6-3 by Nathalie Dechy of France and ninth seed Katarina Srebotnik was upended 4-6 6-4 6-3 by Alona Bondarenko of Ukraine
Fifth seed Francesca Schiavone had no problems, however, beating Italian Roberta Vinci 6-2 6-1 to reach the last eight.
