Second seed Serena, struggling with a virus, lost 7-5, 6-4 to Austria's Sybille Bammer while in-form Italian Flavia Pennetta ended the hopes of third seed Venus with a 7-6, 6-4 victory.
Former world number one Clijsters, unseeded here in her first tournament after a two-year break, won an enthralling two-hour encounter with Russian Kuznetsova.
"Physically I've been training harder than ever," the Belgian told reporters. "I didn't doubt myself [in the third set].
"I have to say I feel really good," added Clijsters, who gave birth to her first child during her time away from the circuit.
"I don't feel tired or drained or anything. Just a few little aches and pains from getting back into playing matches."
Retrieving everything, Clijsters broke in the second game of the match and though Kuznetsova clawed back to 4-4, she snatched the set when the sixth-seeded Russian netted a backhand.
The 26-year-old looked set to win in straight sets when she led 3-0 in the second, but Kuznetsova found form to level at 3-3.
HEAVY GROUNDSTROKES
The Russian denied Clijsters a 5-3 lead by correctly challenging a call, then broke the Belgian in the following game and served out the set.
Trading heavy groundstrokes, Kuznetsova saved a break point in the fourth game of the decider before Clijsters stepped up another level and broke twice to seal victory.
Serena Williams was a shadow of the player who won Wimbledon last month and Bammer took full advantage to clinch a meeting with fifth seed Jelena Jankovic, who edged out Victoria Azarenka 7-5, 7-6.
Sister Venus racked up 38 unforced errors as 14th seed Pennetta outplayed the American to clinch her fourth victory in their seven meetings.
"I'm really happy," Pennetta said. "After winning last week in Los Angeles I took a lot of confidence and it showed on the court today."
Pennetta, who has now won 14 consecutive matches, constantly made Williams play one more shot than she wanted and unleashed a series of stunning passing shots.
The Italian won the tiebreak 7-2 and one break in the second set was enough to give her victory.
"After a while it was obvious that she was just keeping the ball in play and waiting for me to self-destruct," Williams said.
"I kept pressing that red button today and it didn't work out."
Top seed Dinara Safina will be Clijsters' next opponent after she easily beat Peng Shuai of China 6-3, 6-4.
Pennetta will play unseeded Daniela Hantuchova after the Slovakian beat seventh seed Vera Zvonareva of Russia 7-6, 0-6, 7-6 in a thrilling battle lasting two hours and 50 minutes.
Fourth seed Elena Dementieva of Russia will meet Caroline Wozniacki in the other quarter-final.