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Former world No 1 Williams, who has won seven Grand Slam events but is rebuilding her game and her life after being diagnosed with an autoimmune disease, got into her stride only at the end of a one-sided match.
Radwanska dominated the court and went 5-1 up in the second set before Williams began to fight back.
The American, watched by sister Serena who made a shock first-round exit on Tuesday, won the next game to love then broke for 5-3.
With light rain beginning to fall on the Philippe Chatrier court, Williams was one point from winning the next game but Radwanska got to matchpoint when she scooped the ball over the American's head, leaving her standing.
A forehand mistake by Williams finished off the match in exactly an hour.
Latest results from the fourth day of the 2012 French Open at Roland Garros on Wednesday (x denotes seeded player):
Men
Second round
Novak Djokovic (SRB x1) bt Blaz Kavcic (SLO) 6-0, 6-4, 6-4
Nicolas Devilder (FRA) bt Michael Berrer (GER) 7-6 (7/5), 6-4, 6-2
Andreas Seppi (ITA x22) bt Mikhail Kukushkin (KAZ) 6-7 (5/7), 6-2, 2-6, 6-1, 6-2
Fernando Verdasco (ESP x14) bt Gilles Muller (LUX) 6-7 (5/7), 6-3, 6-2, 6-2
Gilles Simon (FRA x11) bt Brian Baker (USA) 6-4, 6-1, 6-7 (4/7), 1-6, 6-0
Stanislas Wawrinka (SWI x18) bt Pablo Andjar (ESP) 7-6 (7/3), 6-7 (4/7), 6-2, 6-1
Roger Federer (SWI x3) bt Adrian Ungur (ROM) 6-3, 6-2, 6-7 (6/8), 6-3
Nicolas Mahut (FRA) bt Martin Klizan (SVK) 4-6, 6-4, 7-6 (7/5), 6-3
Lukasz Kubot (POL) bt Florent Serra (FRA) 7-6 (7/0), 6-2, 7-6 (7/4)
Juan Martn Del Potro (ARG x9) bt Edouard Roger-Vasselin (FRA) 6-7 (5/7), 7-6 (7/3), 6-4, 6-4
Marin Cilic (CRO x21) bt Juan Carlos Ferrero (ESP) 7-6 (7/4), 6-2, 6-3
Kevin Anderson (RSA x31) bt Horacio Zeballos (ARG) 6-3, 3-6, 6-3, 6-0
Tomas Berdych (CZE x7) bt Michael Llodra (FRA) 6-2, 6-3, 6-3
Women
Second round
Victoria Azarenka (BLR x1) bt Dinah Pfizenmaier (GER) 6-1, 6-1
Aleksandra Wozniak (CAN) bt Zheng Jie (CHN x31) 6-2, 6-4
Mara Jos Martnez (ESP) bt Lucie Safarova (CZE x20) 6-4, 7-5
Dominika Cibulkova (SVK x15) bt Vania King (USA) 6-0, 6-2
Sloane Stephens (USA) bt Bethanie Mattek-Sands (USA) 6-1, 6-1
Mathilde Johansson (FRA) bt Petra Cetkovsk (CZE x24) 7-6 (7/1), 6-2
Nadia Petrova (RUS x27) bt Chanelle Scheepers (RSA) 6-3, 6-3
Samantha Stosur (AUS x6) bt Irina Falconi (USA) 6-1, 6-4
Agnieszka Radwanska (POL x3) bt Venus Williams (USA) 6-2, 6-3
Svetlana Kuznetsova (RUS x26) bt Chan Yung-Jan (TPE) 6-4, 7-6 (7/4)
Sara Errani (ITA x21) bt Melanie Oudin (USA) 6-2, 6-3
Ana Ivanovic (SRB x13) bt Shahar Peer (ISR) 6-2, 6-2
Flavia Pennetta (ITA x18) bt Alexa Glatch (USA) 6-3, 6-1
Anabel Medina Garrigues (ESP x29) bt Irena Pavlovic (FRA) 6-3, 6-2
Petra Martic (CRO) bt Marion Bartoli (FRA x8) 6-2, 3-6, 6-3
