The Pole, who will next face Czech 30th seed Barbora Strycova, broke six times as her French opponent looked nothing like the player who won the Strasbourg International last Saturday.
Garcia won a higher percentage of points on her second than on her first serve, which was all too easy to read for Radwanska, who reached the quarter-finals at Roland Garros in 2013.
Garcia threatened a comeback in the second set after falling 3-0 down, but two successive double faults handed Radwanska her third match point, which she converted as the Frenchwoman mishit a backhand.
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(Reporting by Julien Pretot; Editing by Toby Davis)
