She will face France's former Wimbledon finalist Marion Bartoli, who beat Yanina Wickmayer 6-4, 7-5.
Meanwhile, Marion Bartoli needed only eight games to beat Jelena Jankovic, who retired feeling unwell.
Both Bartoli and Jankovic were already out of contention to reach the last four.
Cavaday's reward for beating Bardina is a meeting with fifth seed Marion Bartoli of France.
Bartoli, meanwhile, reached her first WTA final in two years after thrashing Belgium's Yanina Wickmayer.
Ivanovic, the 2008 Indian Wells winner, beat French seventh seed Marion Bartoli 6-3 6-4 in their quarterfinal.
Bartoli, the famous mezzo soprano is as much a force of nature as she is a singer.
Ms Bartoli focuses on the music and influence of Nicola Porpora, a largely forgotten 18th-century Neapolitan musician.
She next plays unseeded Francesca Schiavone, the 2010 champion, who beat No. 13 Marion Bartoli of France 6-2, 6-1.
At the WTA event in Paris, France's Marion Bartoli will face Germany's Angelique Kerber in the final on Sunday.
The Williams sisters Venus and Serena are not playing, nor is Bank of the West Classic winner Marion Bartoli.
But Bartoli evened it up at 4-4 before a further break of service saw Clijsters one set to the good.
Seventh seed Marion Bartoli awaits Ivanovic in the last eight, following the Frenchwoman's win over Czech No. 23 Lucie Safarova.
Kerber took the opener and led 3-0 in the second set until Bartoli battled back in typical style to level.
As a mezzo, Ms Bartoli cannot match Farinelli's reportedly celestial tone, nor is she in possession of his man-sized lungs.
The Italian, also the 2011 runner-up but now dropped to 50th in the rankings, crushed French 13th seed Marion Bartoli 6-2 6-1.
Bartoli, the 2007 Wimbledon runner-up, was hampered by a hip injury during the match, but still managed to defiantly complete the match.
Kvitova's win sets up a match against another French opponent, Marion Bartoli.
Bartoli will next face Azarenka, who believes she can regain a place among the world's top 10 after struggling in recent months.
BBC: Marion Bartoli defeats Ana Ivanovic at Stanford Classic
Bartoli, who received a wild-card invite after making a late entry, will face former world No. 1 Caroline Wozniacki in the quarterfinals.
Bartoli, who recently won the Stanford title beating Venus Williams in the final, said her Belgian opponent showed little sign of her layoff.
The Australian lost in straight sets, 6-4 6-2, to American Laura Granville, with ninth seed Marion Bartoli losing 6-3 6-0 to Thai Tamarine Tanasugarn.
Best vocal performance was one of the strongest categories this year, but a disc called OPERA PROBITA from Cecilia Bartoli was easily my favorite.
However, 15th seed Marion Bartoli was the second seed to fall on Wednesday when she was beaten 3-6 6-3 6-0 by Russia's Vesna Manasieva .
Elsewhere, Mario Bartoli , the French No. 15 seed, recovered from an injury to her ankle as she beat Spaniard Maria-Teresa Torro-Flor 6-4 2-6 6-4.
Bartoli's win, in her final group match, came too late to keep her in the tournament and Jankovic's retirement means she was denied a proper victory.
Meanwhile, Marion Bartoli of France beat Anabel Medina Garrigues of Spain 6-2 6-1 and seventh-seeded Marlene Weingartner of Germany eased past Aranxta Parra of Spain 6-2 7-5.
Williams secured an early break in the first game of the first set before Bartoli pegged her back but the number three seed closed out the opener 6-3.
The 22-year-old won 7-5 7-6 (7-4) against the 2012 French Open runner-up as she followed up earlier victories over third seed Marion Bartoli and No. 5 Roberta Vinci.
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