Ostapenko, though, produced the required resolve when play resumed, with a double break putting her on the brink of victory, which she delivered when holding her serve to love for a 4-6 6-2 6-2 win over the 11th seed in just under two hours on court.
It will come as a real blow to former world No 1 Wozniacki, who was chasing a first Grand Slam title and started strongly in a game that was scheduled to start at 1pm local time but didn't finish until close to 8pm due to a postponed start and regular rain breaks.
"Lucky her, she's way younger than I am", said Bacsinszky, the No. 30 seed from Switzerland who defeated France's 13th-seeded Kristina Mladenovic 6-4, 6-4 on Tuesday in a quarterfinal match twice delayed by persistent rain and stormy weather for a total of nearly four hours.
Logic suggested the steady approach she adopted after losing the first-set tiebreak would get the job done for the experienced Swiss, not the white-knuckle rollercoaster tennis Ostapenko was playing.
French hopes now rest with Caroline Garcia, who faces world number three Karolina Pliskova today, while Elina Svitolina of the Ukraine plays Romania number three seed Simona Halep. Bacsinszky didn't back down, fighting it off on an aggressive passage of play before bringing up match point with a perfectly placed victor down the line. Ostapenko had a little more trouble with Carolina Wozniacki, winning 4-6, 6-2, 6-2.
Both players were gleaming with joy after the match as they face what could be the best gift or a birthday heartbreak on Thursday. Ostapenko is going for her shots, and it's worked up to this point.
The 19-year-old bounced back from some sloppy play in the first set and dealt with the rain delays far better than her more experienced opponent.
But at last year's US Open, she lost in the Final, and she made it to the quarterfinals of the Australian Open to start the 2017 Grand Slam season. She is also the first teenager to go past the fourth round since Ana Ivanovic did so at the 2007 French Open.
"I feel extremely privileged to be in a second semi-final at Roland Garros", said Bacsinszky, whose run to the last four two years ago was ended by Serena Williams in three sets.
Wozniacki merely had to keep the ball in play to build a 5-0 lead as her unseeded opponent sprayed errors. She's had great results here in the past. In the end of the day, I'm just looking at it in this certain match.
"I have to take it one step at a time and just - I'm still in the semi-final, and it's going to be a hard match".
Catherine Bellis then took her to three sets, before Svetlana Kuznetsova did the same.
Because of showers and heavy wind much of the afternoon, the two men's singles matches on Tuesday's schedule have been moved to Wednesday: No. 4 Nadal vs.