Safina, the top seed and last year´s runner-up to Ivanovic, cruised to a 6-2, 6-0 victory over fellow Russian Anastasia Pavlyuchenkova in just 67 minutes, while Ivanovic, seeded eighth this year, needed a mere 61 minutes in a 6-0, 6-2 rout of the Czech Republic´s Iveta Benesova.
Williams, meanwhile, was overmatched by the 29th-seeded Szavay, who posted a 6-0, 6-4 triumph on Court Lenglen. Szavay has now bettered her two previous performances at the French Open -- a second-round result in her 2007 debut and a third-round exit last year.
The third-seeded Williams, however, continued her run of frustration on the famed red clay in Paris. She has played the French Open every year since a second-round setback in her Grand Slam debut at Roland Garros in 1997 and has now been sent home in the third round for a third consecutive year.
Only once has the former world No. 1 reached the final of this Grand Slam event, losing to her sister Serena in the 2002 title match. She will again have to look forward to her favorite major tournament, Wimbledon, where she is the two-time defending champion and has won five of her seven career Grand Slam titles.
Slovakia´s Dominika Cibulkova, the 20th seed, also advanced to the fourth round with a 6-4, 6-2 victory over Argentina´s Gisela Dulko and France´s Aravane Rezai toppled Portugal´s Michelle Larcher De Brito, 7-6 (7-3), 6-2.
Spain´s eighth seed Fernando Verdasco became the first man into the fourth round after beating compatriot Nicolas Almagro in straight sets.
Verdasco did not have things all his own way during a 6-2 7-6 (7-4) 7-6 (10-8) victory but his greater consistency was the difference and, after nine match points, he prevailed to set up a clash with 10th seed Nicolay Davydenko.
Davydenko, a two-time semi-finalist at Roland Garros, saw off Stanislas Wawrinka 6-3 4-6 6-3 6-2.
Fourth seed Novak Djokovic took just 35 minutes to finish his match with Sergiy Stakhovsky and reach the third round of the French Open with a 6-3 6-4 6-1 win.
Djokovic, who led by two sets when darkness stopped play on Thursday, kept his momentum when the match resumed on Friday to outplay the world number 110.