Gauff, Medvedev through in Australia as Djokovic begins record Slam quest
A bumper day two at the season-opening Grand Slam also sees six-time major winner Iga Swiatek, home hope Alex de Minaur and fourth seed Amanda Anisimova enter the fray.
But the tournament is over for seventh seed Felix Auger-Aliassime who retired from his opening match with cramp.
Gauff came through against Uzbekistan's Kamilla Rakhimova on centre court 6-2, 6-3, but was less than convincing, with seven double faults and 31 unforced errors.
"I tried not to put too much pressure on myself," she said. "I am only satisfied if I win, but I am proud of myself regardless of how I get on."
Her reward is a clash next with Venus Williams' conqueror Olga Danilovic.
Gauff last year beat world number one Aryna Sabalenka at the French Open for her second major crown, but her best performance on the Melbourne hardcourts was a semi-final in 2024.
While she progressed, 2020 winner Sofia Kenin crashed out 6-3, 6-2 to fellow American Peyton Stearns as temperatures soared past 30C and fans again faced long queues to get into Melbourne Park.
Canada's Auger-Aliassime was another casualty, calling for the trainer while down 3-6, 6-4, 6-4 to Portugal's Nuno Borges and pulling out.
"I want to be on the court competing with my opponent," said Auger-Aliassime. "I don't want to be just standing there, you know, like a punching bag."
Russia's 11th-seeded Medvedev, runner-up in 2021, 2022 and 2024, warmed up for Melbourne with victory in Brisbane and ground to a 7-5, 6-2, 7-6 (7/2) opening win over Dutchman Jesper de Jong after twice being broken when serving for the match.
"The most important is to win," he said.
Men's 13th seed Andrey Rublev also progressed, but women's 15th seed Emma Navarro tumbled out.
'Like my chances'
Serbian great Djokovic, 38, faces Spain's 71st-ranked Pedro Martinez in the final match of the day on Rod Laver Arena.
A former world number one, now ranked four, Djokovic is the undisputed king of Melbourne, having won a record 10 Australian Open crowns.
He owns 24 major titles, equal for the most ever with Australia's Margaret Court, but a 25th has remained agonisingly out of reach.
With Carlos Alcaraz and Jannik Sinner now dominant, Djokovic's last Grand Slam victory came at the US Open in 2023.
Despite age and injury catching up with him, Djokovic said on the eve of his favourite tournament: "I know that when I'm healthy, when I'm able to put all the pieces of the puzzle together on a given day, I feel like I can beat anybody."
He added: "I like my chances always in any tournament, particularly here."
Australia's De Minaur, the sixth seed, will have the Rod Laver Arena crowd roaring him on against 113th-ranked Mackenzie McDonald of the United States.
Poland's Swiatek, the second seed, has never gone beyond the last four in Melbourne.
Like Alcaraz, she is pursuing a career Grand Slam of all four major titles, having triumphed previously at Wimbledon, the US Open and French Open.
Swiatek plays Chinese qualifier Yuan Yue while the American Anisimova, runner-up last year at Wimbledon and the US Open, meets Switzerland's Simona Waltert.
Other notables in action include the 2014 champion Stan Wawrinka, who was handed a wildcard aged 40 in his last Australian Open before retirement.