Rybakina, who will meet Victoria Azarenka for a place in the final, needed nearly three hours to beat Sakkari.
The Greek saved two match points in the second set before ultimately succumbing to the fourth seed's overwhelming serve in the final game of the third set.
"I have no words because I'm so tired," Kazakh Rybakina said.
"It was such a tough battle. I got a little bit lucky with new balls, so it was easier to serve it out ... now, I just need to recover."
Rybakina, the world No.4 and 2022 Wimbledon champion, will need to bring her best again when she takes on three-time tournament champion and former world No.1 Azarenka.
The veteran Belarusian player contended with a near hour-long stoppage in the first set because of a power outage to the electronic line-calling system before beating Yulia Putintseva 7-6 (7-4) 1-6 6-3 earlier in the day.
Azarenka saved three set points en route to clinching the first set tiebreak in the 90-minute opener before her unseeded opponent raised her game in the second to force a decider.
Azarenka proved too solid down the stretch, firing an unreturnable serve on match point.
"Very happy with today's win," the 34-year-old told reporters.
"It's good to be back in the later stages of the tournament, it's what I work for."
