Fourth seed Clijsters, who made her return to tennis at this event a year ago before winning her second Grand Slam title a month later at the US Open, ousted Italy's Flavia Pennetta 7-6 (8/6), 6-4.
Clijsters fired seven aces and connected on 67 percent of her first serves but the Italian would not go out easily, saving 11 of 14 break-point chances by Clijsters.
Ivanovic dominated without facing a break point in a 6-1, 6-3 triumph over Akgul Amanmuradova of Uzbekistan that lasted just 56 minutes.
Russian teen Anastasia Pavlyuchenkova recovered from a 5-1 deficit in the first set and outlasted Belgium's Yanina Wickmayer 7-5, 3-6, 6-1.
It was the ninth match triumph in a row for the Russian, who is coming off a title at Istanbul.
Russian star Maria Sharapova and France's Marion Bartoli meet in a later quarter-final for the chance to meet Pavlyuchenova in Saturday's other semi-final.