Ivanovic, the third seed, beat Kurumi Nara of Japan 6-2 6-3 in 76 minutes.
She was made to work harder than her second-round match, when she romped past Johanna Larsson 6-1 6-1, but after a few early wobbles found her range to control the contest.
Her forehand, in particular, was working well and she rattled off a number of driving winners.
Ivanovic will take on her good friend Flipkens in the semifinals after Flipkens beat qualifier Sachie Ishizu 6-4 7-5 in the day's opening quarter-final.
It was the third seed's third straight sets victory of the tournament but she looked in trouble when trailing 5-2 in the second set. She then broke Ishizu twice to take the set and match in one hour 24 minutes. Flipkens was a beaten quarter-finalist at last year's ASB Classic.
Ivanovic and Flipkens are due to play their doubles quarter-final later today against third seeds Lucie Hradecka and Michaella Krajicek.
Fifth seed Jamie Hampton and Lauren Davis will meet in an all-American contest before Venus Williams takes on Spain's Garbine Muguruza in the final singles quarter-final of the day.
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