The Serb, who lifted the Suzanne Lenglen Cup in 2008, shrugged off the windy conditions to set up a meeting with Czech Lucie Safarova.
Ivanovic arrived at the tournament on the back of a mediocre claycourt season, losing in the first round in Stuttgart, the third round in Madrid and the second round in Rome.
"You have to just find it within yourself and trust in the work you have done. You have to trust in your team and execution on the court, because sometimes you would give your best out there and it won't be enough," the former world number one told a news conference.
"You have to accept that and not dwell on it and then lose more matches in that manner."
She made sure she would not lose again on Tuesday, taking control early on and never allowing her 19th-seeded opponent into the contest on Court Philippe Chatrier.
She put an end to a one-sided encounter on her third match point with a forehand passing shot to reach the last four of a grand slam for the first time since 2008.
"I don't know if I should feel old or happy," the 27-year-old told a courtside interviewer.
"The only thing I could do was stay calm. The ball was going all over the place."
Her powerful forehand earned Ivanovic the first break as she went 2-0 up only for the Serb, under the watchful eye of her boyfriend, Germany midfielder Bastian Schweinsteiger, to drop serve in the following game.
She then broke again for 3-1.
The sun crept through and the spectators came back from their late lunch to watch Ivanovic clinch the opener by holding serve to love.
Ball girls and boys wiped their teary eyes as strong winds sent the red dirt swirling at the change of ends before Ivanovic resumed her domination.
She stole Svitolina's serve in the first game and saw off two break points en route to holding for 2-0. She broke again for 5-2 and finished it off on her serve.
Looking back on her earlier triumphs, Ivanovic said she did not appreciate her successes like she does now.
"I just was so excited and I felt like I could achieve anything. At that point I had so many victories and so many good results," she said
"Maybe I didn't appreciate it as much as I do now."
(Editing by Tony Jimenez and Toby Davis)
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