The Belarusian's 7-6 (7-2) 4-6 6-3 victory on Friday keeps her on track to end a two-year title drought.
Azarenka kept her poise under a barrage of powerful serves as the world No.7 lobbed 10 aces across the net.
She broke down Gauff's stout defence to convert on five of 13 break point chances and set up a meeting with either Jessica Pegula or Sloane Stephens of the United States.
"She served really well, she was mixing a lot so I was trying to adjust," Azarenka said. "I tried to just stay consistent."
Gauff, who reached her first grand slam final at Roland Garros earlier this year, had already booked her spot at the WTA Finals and the last qualification spot will be decided in a face off on Friday between Maria Sakkari and Veronika Kudermetova.
"She's the same age as my career so it felt pretty interesting," said Azarenka. "She's such a great athlete, I can't wait to see how far she can go."
The WTA Finals, featuring the world's top eight singles players and doubles teams, will start in Fort Worth, Texas on October 31.
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