Venus, seeded 30th here, has been struggling with a back problem of late on top of Sjogren's Syndrome but she certainly has not lost her fighting spirit.
In a match that featured 21 breaks of serve, 32-year-old Venus recovered from 4-0 down in the second-set tie-break and then 5-1 in the decider before Radwanska clinched victory on her second match point with the light fading.
A troublesome back injury has limited Williams to two matches over the previous 18 months, preparation she herself described as "extremely unideal" to the Associated Press.
"I can't really serve very hard. It's painful when I do that. But I'm getting better. I just, you know, ran out of time to get better for this tournament," Williams said.
"My strategy was more or less to put the ball in, and that's very difficult for me too, because that's not who I am. But that's all I had."
Urszula is the younger sister of Agnieszka Radwanska, the same player who defeated Williams in the second round at Roland Garros a year ago.
"Yeah, of course, I was talking with Aga about Venus," Urszula said.
"I was well-prepared for this match, and I knew she was a great fighter, so I should be focused the whole match."
Williams was broken 11 of the 17 times she served Sunday.