Ivanovic looked anything but the world number seven as she lost the opening five games, prompting her coach Dejan Petrovic to join her in an animated courtside chat.
Kanepi double faulted when she had the chance to wrap up the set 6-0 then Ivanovic responded by breaking her serve and reeling off the next four games.
Kanepi regained her composure to eventually take the set but the momentum was with Ivanovic and the Serbian ran away with the last two sets to win 4-6 6-4 6-3.
"Definitely I had very slow start. I felt like I was hitting the ball all right, but I was just missing the easy put-aways," Ivanovic said.
"I nearly came back in the first set, but was really happy I kept my composure for the second and definitely in the third to stay on top."
Ivanovic will now play American Varvara Lepchenko in Friday's semi-finals after she beat Alla Kudryavtseva of Russia 7-5 7-5.
Sharapova was very impressive in her 6-1 6-3 win over Spain's Carla Suarez Navarro. The world number two has given up just five games in her two matches so far but said she is not worried about having too easy a workout leading into the Australian Open.
"If I wanted to I could make it tougher on myself. I don't think that's really the goal," she said.
"I know how to get ready for grand slams. I'm certainly not thinking about that I need that type of match. My goal is to go out there and do the right things."
The Russian will now play Elina Svitolina in the semis after the Ukrainian upset third-seed Angelique Kerber of Germany. Svitolina, the 2010 junior French Open champion, was a set and a double break down before she rallied back to win 4-6 7-5 6-3.
"I faced Svitolina last year in Beijing for the first time. She's a young girl that's already in the top 30 with a tremendous amount of potential and someone that you'll be seeing in the top very soon," Sharapova said.
"She has a good game, an all-around player. It'll be a tough matchup, but one that I look forward to actually."
(Reporting by Julian Linden; editing by Justin Palmer)
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