Gauff dominated the resulting tie-break and broke in the first game of the third set before Sakkari regrouped.
The ninth seed immediately rescued the break and won four games in a row to take a 4-1 lead.
Gauff served to stay in the match at 5-2 down but, despite saving another match point, was unable to hold off Sakkari.
“I don’t want to disrespect Coco but it would’ve been amazing if this match was over an hour ago,” Sakkari joked afterwards.
“Coco is such an amazing opponent. I’m just so happy I’m in the final once again here.”
In the men’s draw, defending champion Carlos Alcaraz plays Australian Open champion Jannik Sinner in the last four from 19:00 GMT, followed by last year’s runner-up Daniil Medvedev against home hope Tommy Paul in the second semi-final.
