Meanwhile, Clijsters, the 2005 Miami champion, needed four match points to beat Henin 6-2 6-7 7-6.
The Williams sisters are level with Switzerland's Martina Hingis and Justine Henin of Belgium.
The second set was a complete switch, with Davenport struggling and Henin-Hardenne finding some form.
Safina's defeat follows the first-round exits of big stars Justine Henin, Maria Sharapova and Svetlana Kuznetsova.
Henin will now play either Svetlana Kuznetsova of Alona Bondarenko who play in the other semi-final.
Henin-Hardenne had 16 winners and 18 unforced errors, while Kuznetsova committed 29 unforced errors with 10 winners.
Belgians Kim Clijsters and Justine Henin, seeded one and two for the tournament, both received first-round byes.
Henin, unseeded in the season's first Grand Slam, edged the No. 19 seed Petrova, 7-6 (7-3), 7-5.
Henin will next play fellow Belgian Yanina Wickmayer, who defeated Italy's Sara Errani 6-1 6-7 (4-7) 6-3.
Henin came from behind to oust second seed Caroline Wozniacki, beating the young Dane 6-7 (5-7) 6-3 6-4.
Henin-Hardenne slid down the world rankings in the second half of 2004 after contracting the illness in April.
"I knew it was going to be tough today and I was ready to fight, " Henin-Hardenne told BBC Sport.
Henin and Clijsters are tied at 12 wins apiece from their 24 meetings.
Justine Henin also picked up a knock in Belgium's 3-2 victory over Estonia.
But the greater experience of 27-year-old Henin proved the difference in the decider.
In the day's final tie Belgium , without Justine Henin-Hardenne and Kim Clijsters, took the lead against Spain .
"I was expecting a tough match because she gave me a lot of trouble in the past, " said Henin.
And Clijsters again dismissed suggestions that she and Henin-Hardenne do not get on.
Mr Melenchon says he will stand in the working-class town of Henin-Beaumont, near Calais, where Ms Le Pen resides.
Kuznetsova then had three game points on her service, but Henin-Hardenne recovered to win the first of six straight games.
Since Justine Henin won her last title in 2007, there have been five different French Open champions in five years.
Davenport's next opponent will be Justine Henin , the Belgian fourth seed who defeated Marie-Gaianeh Mikaelian of Switzerland, 7-5, 6-2.
"Somebody like Justine Henin is playing good tennis but in terms of marketability she's not quite up there, " he says.
Belgian ace Henin crashed out to Aravane Rezai, losing eight straight games to fall to a 4-6 7-5 6-0 defeat.
Henin's fellow-returning Belgian Kim Clijsters hardly needed to raise a sweat as she beat Canadian qualifier Valerie Tetreault 6-0 6-4.
Sharapova's victory means she will meet her compatriot Anna Chakvetadze in the semi-finals, while Ivanovic plays defending champion Justine Henin.
"She has confidence and wants to be in the final, so it's going to be a great match, " said Henin-Hardenne.
Henin-Hardenne twice broke in the first set, only to be pegged back, but on the third occasion, Bremond could not recover.
"All the things that happened this week, including my injury, Kim's injury -- I've not had the best preparation, " Henin said.
Williams will now meet Kim Clijsters in the final, after the Belgian saw off compatriot Justine Henin in three tough sets.
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