The reigning US Open champion needed four sets and a large dose of composure to see off Kyrgios and silence Rod Laver Arena, but it was altogether more comfortable as he dismissed unseeded Dutchman Botic van de Zandschulp.
The 25-year-old Medvedev kept his cool in sweltering conditions on Margaret Court Arena to roll through 6-4, 6-4, 6-2 in 1hr 55min and faces American Maxime Cressy in the fourth round.
"I'll put it this way, it's easier to play a guy from the Netherlands than a guy from Australia in Australia in Melbourne," said Medvedev, who has a similar love-hate dynamic with the crowd in New York.
"Every good relationship must have its ups and downs so I think it's good, it's entertaining and it's real, there is some relationship going on."
Greek fourth seed Tsitsipas, who is chasing a maiden major, was also largely untroubled in a 6-3, 7-5, 6-7 (2/7), 6-4 win over unseeded Frenchman Benoit Paire and plays American Taylor Fritz next.
Tsitsipas's serve was the foundation. He did not drop a service game despite losing a set and sent down 21 aces.
Medvedev and Tsitsipas, two of the generation coming up fast behind the "Big Three" of Djokovic, Rafael Nadal and Roger Federer, could meet in the semi-finals.
Felix Auger-Aliassime sent home the last Briton left standing, the 21-year-old Canadian thrashing Dan Evans 6-4, 6-1, 6-1.
There is an intriguing clash later Saturday between fifth seed Andrey Rublev of Russia and 2018 Melbourne Park finalist Marin Cilic of Croatia that has the potential to be a late-night classic on Margaret Court Arena.
Cilic pushed Roger Federer -- who is injured and not in Melbourne -- to five sets in the final four years ago and the big-serving 27th seed is capable of upsetting anyone.
'What else can I ask for?'
In the women's draw, Halep swatted aside Danka Kovinic -- conqueror of US Open champion Emma Raducanu -- 6-2, 6-1 in just 64 minutes to set up a date with Cornet of France.
The fit-again Halep came into the tournament full of confidence after her first title in 16 months earlier this month at a Melbourne warm-up event and was always in charge.
"I feel great that I can play the fourth round again. I always love to play in Australia so that's why maybe I play good matches," said Halep, seeded 14.
Unseeded Cornet celebrated her 32nd birthday by slugging it out with Slovenia's 29th seed Tamara Zidansek for a 4-6, 6-4, 6-2 victory.
The crowd serenaded her by singing happy birthday.
"It's definitely a very, very special victory being back in the second week 13 years after my first (time). It's quite special," she said.
"The day of my birthday, I mean, what else can I ask for?"
World number two Sabalenka overcame the serving problems that have been plaguing her early season, but still needed three sets to get past 31st seed Marketa Vondrousova.
The Belarusian had served 70 double faults in her previous four matches in 2022, but only tallied 10 as she beat the Czech 4-6, 6-3, 6-1.
She now meets unseeded Estonian veteran Kaia Kanepi.