Bacsinszky downed French hope Kristina Mladenovic 6-4, 6-4 on a rain-hit day at Roland Garros, while 19-year-old Ostapenko stunned Caroline Wozniacki 4-6, 6-2, 6-2.
Bacsinszky or Ostapenko can make it a double celebration when they meet on Thursday as both women celebrate their birthdays.
Bacsinszky will be 28; Ostapenko will turn 20.
Tuesday's matches both lasted just under two hours in duration but took the best part of six hours to complete after two rain delays — one lasting three hours.
“I feel extremely privileged to be in a second semifinal at Roland Garros,” said Bacsinszky, whose run to the last four two years ago was ended by Serena Williams in three sets.
“It was extremely difficult today. There was fine weather, then rain, wind and a sandstorm.”
“I had to stay concentrated. I am sorry for the crowd but happy for me.”
Mladenovic, bidding to be the first Frenchwoman in the semifinals since Marion Bartoli in 2011, was broken in her first service game by Bacsinszky before she levelled at 2-2.
But she handed the advantage straight back in the fifth game and the Swiss player went on to claim the opening set on an ace after 63 minutes on court.
At 1-1 in the second set, the match was halted for three hours as torrential rain swept across Roland Garros.
When they returned, Mladenovic raced into a 3-2 lead.
She was again unable to back it up and Bacsinszky took the next four games to stretch to what proved to be a crucial 5-3 lead.
Ostapenko became the first Latvian woman to reach the last four of a Grand Slam after toppling former world number one Wozniacki.
The Danish 11th seed charged into a 5-0 lead in the opening set, blowing two set points as Ostapenko reeled off four straight games before Wozniacki finally wrapped it up.
But the Latvian, who is a chasing a first career title, hit back impressively with three breaks of serve in the second set to level the match.
Wozniacki hit just six winners in total, and Ostapenko's bold approach paid off as the teenager continued her dream run.
“I'm really happy, I cannot believe it still,” said Ostapenko.
“I had to stay aggressive. I was missing sometimes but in the end it worked out. It will be really great to play here when it will be my birthday. I hope I can enjoy it.”