Mladenovic was hoping to be the first French to win at Rolland Garros since Mayy Pierce in 2000 but was completely outclassed by her Swiss opponent.
Ostapenko wasted no time in wrapping up the set when play resumed after a long wait due to the weather and Wozniacki saved a break point to take a 2-1 lead in the decider before the heavens opened again. Two more quarterfinals matches featuring all seeded players take place on Wednesday at Roland Garros.
The men's quarter-finals featuring nine-time champion Rafael Nadal and holder Novak Djokovic were postponed until Wednesday after two rain delays disrupted Tuesday's play. Bacsinszky defeated 13th-seeded Kristina Mladenovic 6-4, 6-4 in a match twice interrupted by rain.
19-year-old Latvian Jelena Ostapenko in the first set of the match against an experienced Caroline Wozniacki has conceded with the account 0:5, but nearly saved the party. "I am going to enjoy it, but I will have (birthday celebration) plans after the tournament".
Following the match, Ostapenko, who fired 38 winners and improved to 4-0 lifetime against Wozniacki, spoke about a number of topic on her run in Paris: "Every match we played was a very tough match, because she's playing really well, and she's very consistent".
It is windy on both courts and dust from the clay is getting in players' eyes.
In fact, one of them will reach their first Grand Slam final, with neither having gone beyond the semi-final stage and, incredibly, Ostapenko had never been beyond the second round before the start of 2017! I started off really, really well.
It was Bacsinszky's 14th victory in her last 16 matches in Paris.
And to keep things nice and simple the victor of Mladenovic/Bacsinszky will meet the victor of Ostapenko/Wozniacki in the semi-finals, likewise the winners of today's men's quarters.
"I'm totally comfortable with what I did and said during the Fed Cup", said Bacsinszky, seeded 30th at this year's French Open. Waves of weather - "We had a hurricane, a sandstorm, and we nearly had snow too", Bacsinszky said in her on-court interview - disrupted both players' games, but she said she was ready for a heavy surface.
The unseeded surprise package was rewarded for a positive approach on Court Suzanne Lenglen and can look forward to the biggest match of her career against Timea Bacsinszky on Thursday, when she will turn 20.
"It's pretty amusing", Bacsinszky, soon to be 28, told reporters.
But Ostapenko found her mental resolve and clawed it back to 4-5 before Wozniacki broke to love to take the opener. Disappointed to go out before the quarters, she walked off the court upset, amid boos from the crowd. "Of course, it's good to have conserved your energy, but for Thiem I think I'd have liked to see him answer a few more questions", the Swede, in Paris as a presenter for Eurosport's daily flagship show "Game, Set and Mats", told Reuters.