Title favourite Halep was being overpowered and looked set for a disappointing exit before turning things around to win 3-6 7-6 (8/6) 6-0.
"I asked him if he wants to come back before Madrid". It was the kind of match that she might have given up on a few months ago. I just sat down at 5-2.
After Venus Williams and Svetlana Kuznetsova lost Sunday, the French Open - which is missing Serena Williams, Maria Sharapova and Victoria Azarenka - is guaranteed of producing a first-time grand slam victor.
"When I sat down at 2-5 (in the second set) I just said the match is lost".
Andy Murray, last year's runner-up, is also on Chatrier.
Although an all-French women's final remains a possibility, the last French man in the draw, Gael Monfils, was knocked out to extend Les Bleus men's drought for another year - Yannick Noah was the last victor in 1983.
Wawrinka, who has yet to drop a set in this year's tournament, won 6-3, 6-3, 6-1.
Svitolina has matched her best run at a Grand Slam after making the last eight at the French Open for the second time in three years, but she needed to rally from 5-2 down in the final set to overcome world 290 Petra Martic in round four.
"She played unbelievable tennis in the first set until the end of the second set", Halep told reporters.
Svitolina wasn't the only top seed to have a scare in the fourth round with world No.3 Karolina Pliskova overcaming an very bad start to beat unheralded Veronica Cepede Royg of Paraguay 2-6 6-3 6-4 to reach the quarters for the first time.
None of the quarter-finalists had won a Grand Slam throughout their careers and there will be a new French Open champion crowned on Saturday.
"I just tried to make her move more, to open [up] the court, and it came".
Her improvement in the claycourt major bears similarities with the career of Russia's Maria Sharapova, a self-described "cow on ice" on the red dirt who still managed to lift the Suzanne Lenglen Cup twice.
"I had "Svitolina mode" back", Svitolina said about trailing 2-5 in the third set.
Don't let the seedings fool you: Halep will be a strong favorite going into this match, not the least because few players are as risky in a Grand Slam draw as those who had faced match point in a previous round and lived to tell.
Also on the plus side of Halep's ledger: She's a former finalist, and her ability and willingness to get that proverbial "one more ball" back can pay big dividends against a tall, hard-serving player who prefers one-two-three knockout tennis.