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Medvedev remained calm as a robot in the face of fanatic support for Kyrgios to secure a clinical 7-6(1) 6-4 4-6 6-2 win.
The tennis maverick, Kyrgios, returned to the combustible John Cain Arena and gave raucous fans a night to remember.
The world No. 19 is scheduled to compete in the Sydney Classic later this month before making her debut in the January 17 to 30 Australian Open.
US Open champion Emma Raducanu was named the Newcomer of the Year.
Gauff, who shot to fame as a 15-year-old when she reached the fourth round at Wimbledon in 2019, urged the 18-year-old Raducanu to take time away from her phone.
Unforced errors piled as the night wore on and Raducanu never exuded the on-court joy that was a hallmark of her breakthrough despite a supportive crowd.
The 18-year-old Briton became the first qualifier to win a Grand Slam.
Raducanu, 18, shot to fame this month after she became Britain's first female Grand Slam champion for 44 years.
US Open runner-up Leylah Fernandez jumped 45 spots, climbing from No.73 to No.28.
Raducanu became the first qualifier in the history of the Open Era to win a major title.
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