It was Gauff's fourth straight victory over Sakkari and she closed out the match on Saturday in 1 hour and 16 minutes.
The American's only scare in Toronto came in the second set after comfortably leading 5–1.
Sakkari fought back to cut the deficit to 5–4, but Gauff held her nerve, earned a second match point, and converted it to seal the win.
"I'm happy with how I played today. Maria is a tough opponent," Gauff said.
"Her level definitely raised toward the end of that second set. But I'm glad I was able to close it out. That last game was a big test for me."
Gauff will now face Alina Korneeva in the round of 16.
It was a different story for Swiatek, the No.7 seed, however.
Nothing went right for the Polish six-time Grand Slam champion in the first set, when she won just one point out of 17 behind her serve and was broken four times.
Kostyuk, the No.10 seed, won the other 16 on the way to taking it 6-3.
But Swiatek bounced back to win the next two sets comfortably for a 3-6 6-1 6-2 victory.
Swiatek will face Diana Shnaider, who knocked out No.3 seed Jessica Pegula, in the Toronto quarterfinal.
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