The Belorussian, who lost to Serena in the Wimbledon semifinals, will next face Maria Jose Martinez Sanchez of Spain.
Defending champion Serena snatched the first set against Maria Jose Martinez Sanchez and then survived a second-set fightback before prevailing 6-3 7-5.
Thursday's quarterfinals see Serbian sixth seed Jelena Jankovic taking on Russia's Alisa Kleybanova and Australian eighth seed Samantha Stosur playing Spain's Maria Jose Martinez Sanchez.
CNN: Radwanska to face friend Wozniacki in Indian Wells semis
On the other side of the draw, Danish second seed Wozniacki suffered a 6-4 6-2 defeat by Spain's Maria Jose Martinez Sanchez in her third-round tie.
In Sunday's action in the Hopman Cup , Tommy Robredo and Maria Jose Martinez Sanchez gave Spain a 3-0 win over the United States in Perth.
However, there were some shock results on the first day at Roland Garros, with defeats for 10th seed Victoria Azarenka and Spanish 20th seed Maria Jose Martinez Sanchez.
Robson began the year by playing alongside Andy Murray in the Hopman Cup in Perth, where she beat a top-30 player, Maria Jose Martinez Sanchez, for the first time.
Serena had endured a coughing fit and made contorversial accusations of cheating after her three-set victory over Maria Jose Martinez Sanchez on Saturday, but this was a 53-minute stroll.
The host nation needed to win all three rubbers in straight sets to snatch a final spot off the unbeaten Spanish pairing of Maria Jose Martinez Sanchez and Tommy Robredo.
Eighth seed Samantha Stosur beat Argentina's Gisela Dulko 7-6 (7-3) 7-5, and the Australian will face Maria Jose Martinez Sanchez on Tuesday, with the Spaniard having a bye after beating Jankovic in Saturday's Italian Open final.
Laura Robson, 15, had given Britain hope of a first title in Perth with a shock 6-1 7-6 (8-6) win over Maria Jose Martinez Sanchez, but Robredo then came from behind to triumph 1-6 6-4 6-3 as Murray dropped his first sets of the week.
Russia's Svetlana Kuznetsova , seeded 14, made it safely through to round three after a 7-5 6-3 victory over Spaniard Maria Jose Martinez-Sanchez.
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