Former world number one 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.
(Reporting by Clare Fallon; Editing by Pritha Sarkar)