Williams will now play dual former champion Victoria Azarenka, who romped past Caroline Wozniacki 6-3 6-1.
The two matches were rematches of clashes from the Australian Open earlier this year and the results were the same, as Williams and Azarenka prevailed.
Azarenka received a wild card entry into this tournament as she continues her comeback after an injury-disrupted 2014 and produced a fine display against Wozniacki, hitting 26 winners in a match between two former World No 1???s.
Williams started her match against Radwanska with a break to love and that proved the decisive moment in the opening set but the American wobbled badly in the second and found herself facing a third three-setter in four days.
She then powered into a 5-1 lead in the final set but had to weather a Radwanska fightback before smacking a forehand winner down the line to serve out the match at the second attempt.
Williams will now contest her first Qatar semi-final and she has won all four of her previous matches against Azarenka.
“I just tried to stay focused because she wasn’t going to let up,” said Williams after the 2-hour, 16-minute win over Radwanska.
“Every match (this week) has been three sets, so it’s been really challenging.
“No one gives you a point, you have to fight, fight, fight for each point, and I guess it’ll be more of that tomorrow.”
In the other semi-final, ninth seed Carla Suarez Navarro of Spain will take on Lucie Safarova of the Czech Republic.
Suarez Navarro came from a set down to upset top seed Petra Kvitova 3-6 6-0 6-3 while Safarova easily defeated Andrea Petkovic 6-2 6-1.