Clijsters looked in excellent form as she beat Iveta Benesova in straight sets, suffering none of the problems of top seed and home hope Samantha Stosur, who was beaten by the Czech in the last round.
The Belgian fifth seed, playing in her first tournament since August because of an abdominal injury, ran out a 6-3 6-2 winner to set up a last-four clash with Daniela Hantuchova, who was granted a walkover because of Serena Williams' withdrawal yesterday.
Estonian Kanepi, meanwhile, kept her nerve to claim a 6-1 7-6 (9/7) victory over German second seed Petkovic.
After racing through the first set, she tightened somewhat in the second before clinching it in a nailbiting tie-breaker.
She will next face third seed Francesca Schiavone, who saved two match points to beat Serbia's Jelena Jankovic 5-7 7-6 (7/2) 6-3.
In the men's event second and third seeds Gilles Simon and Alexandr Dolgopolov battled through to set up a semi-final meeting.
Dolgopolov beat sixth seed Radek Stepanek 4-6 6-3 6-3, while Simon was a 7-6 (7/2) 6-4 winner over Santiago Giraldo.
The other two quarter-finals, including top seed Andy Murray against doubles partner Marcos Baghdatis, will take place on Friday.