US Open champion Gauff looked far from her best early in the match as Sakkari drew first blood after a short rain delay by breaking the American third seed in the ninth game, before securing the opening set with ease.
The players were forced off the court for a longer spell as the showers returned at the start of the second set, but Sakkari picked up from where she left off when the action resumed and tightened her grip on the contest with a double break.
But Gauff was not done and the 20-year-old rediscovered her rhythm in the high-octane second set to save three match points and go level at 5-5, eventually forcing a tie break which she dominated to set up a decider.
Sakkari regrouped, reeling off the games in the third set to prevail in the late-night thriller.
Top seed Swiatek earlier produced a near-flawless display in chilly, windy conditions to cruise past Ukraine's Marta Kostyuk 6-2 6-1 and remain on track for her second title of the year.
"For sure I'm happy with the performance. I think it was the cleanest match I played here. The focus was high throughout and I didn't have any doubts," said Swiatek, who has not dropped a set in the Californian desert this year.
"It's already a great tournament."
The 2022 champion made her intentions clear early on with an aggressive start to break for a 2-1 lead and pulled further away when Kostyuk hit a double fault two games later.
The 21-year-old Kostyuk surrendered the opening set in 31 minutes and was put through the wringer again by world number one Swiatek at the start of the second set.
A powerful smash at the net gave Swiatek another break. The four-times Grand Slam champion withstood some pressure from the 31st seed in the next game before building a 3-0 advantage.
Kostyuk appeared to struggle with a left foot problem and a medical timeout only delayed the inevitable, as Swiatek eased to a WTA Tour-leading 19th match win of the season, finishing off the match with a forehand winner.
