Number 17 seed Sakkari was playing in her first grand slam quarter-final but showed no signs of nerves as she closed out a 6-4 6-4 victory against last year's champion in one hour and 35 minutes.
Sakkari will play unseeded Barbora Krejcikova in what will be a first major semi-final appearance for both players.
It was a scrappy start to the contest with both players struggling to establish a rhythm on serve.
Early breaks were exchanged before Sakkari had to save four break points in her next game to avoid another and Swiatek then fended off three in the next.
Sakkari looked a different player from then on as she seized control. At 4-4, a powerful forehand winner gave her two break points and she claimed the first of them.
After a first opportunity to move ahead was not taken as she served for the set, two big serves saved a break point and gave her another chance that was taken with a fine backhand, prompting huge emotion from the Greek.
Sakkari had the momentum and went 2-0 up in the second set, at which point Swiatek called for a medical timeout that resulted in a 10-minute delay.
After emerging with strapping on her right thigh, Swiatek looked more confident on her return, holding confidently before saving two break points in her next service game to remain in contention.
But Sakkari was undeterred, serving superbly and not allowing a single break-point opportunity in the second set, clinching a famous triumph that brought tears to her eyes with her third match point.