Following a delay of the best part of three hours, the teenager closed out the set on her serve to level.
"We had all the seasons rolled into one today", Bacsinszky said. There was fine weather, then rain, wind and the clay was blowing in our eyes.
"I had to stay aggressive, and I found my game", said Ostapenko. "I'm really sorry to have knocked out the penultimate French representation".
Contesting her eighth career Grand Slam event, the 47th-ranked Latvian broke serve seven times toppling former world No. 1 Caroline Wozniacki, 4-6, 6-2, 6-2-her fourth win over Wozniacki in as many meetings.
"Because to play semi-final of Roland Garros on your birthday, I think it's really nice". It will be really great to play here when it will be my birthday.
Bacsinszky nearly dropped her serve but held on and won the match when Mladenovic hit a forehand wide.
Mladenovic endured a rollercoaster tournament filled with ups and downs, and she was very emotional on the court on Tuesday, almost bursting into tears just before the second rain delay. With the third game balanced at break point to Bacsinszky, the rains arrived, and play was halted for three hours. The Swiss won a 22-stroke rally with a backhand smash, Mladenovic came back from two break points with another wonderful rally, and would battle though four deuces and another break point before Bacsinszky fired a cracking backhand victor to edge the break.
STAT OF THE DAY: It will be a double birthday celebration when Swiss Timea Bacsinszky meets Jelena Ostapenko in the semi-finals.
LSU names Scalise honorary CWS coach, will send game ball
It was the 22nd straight victory for No. 1 national seed OSU (55-4), which will continue CWS play at 4:05 p.m. The double-elimination tournament features two brackets, and Sunday's games were the first in the second one.
Catherine Bellis' French Open dream run did not extend beyond the third round but the American teenager exited Roland Garros knowing she has won a whole host of new admirers, including the woman who defeated her, Caroline Wozniacki.
The men's quarterfinals involving Rafael Nadal and Novak Djokovic that had been scheduled for Tuesday were postponed until Wednesday.
No. 13 Kristina Mladenovic of France is playing No. 30 Timea Bacsinszky of Switzerland at Court Philippe Chatrier, while No. 11 Caroline Wozniacki of Denmark faces unseeded 19-year-old Jelena Ostapenko of Latvia at Court Suzanne Lenglen.
One Frenchwoman however does remain in the draw: The player Mladenovic won the doubles title with a year ago in Paris, Caroline Garcia, who will play second-seed Karolina Pliskova in Wednesday's quarterfinals.
Nadal, who turned 31 on Saturday, will next face either Milos Raonic or another Spaniard Pablo Carreno Busta for a place in the semi-finals.
Asked if she is surprised with her exploits, Bacsinszky said: "I amaze myself sometimes with the focus I can have on some important moments during the whole tournament".
"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".