This is the first Grand Slam since the Open era began in 1968 to lose the top two seeds in both the men's and women's draws before the last eight.
There has been a series of surprise results at Melbourne Park, and unseeded Pole Linette got in on the action with a 7-6 (7/3), 6-4 victory over France's fourth-seeded Garcia.
The 45th-ranked Linette faces Karolina Pliskova next after the former world number one romped past China's Zhang Shuai 6-0, 6-4.
Linette gave a sizeable contingent of red-and-white-clad Polish fans at Rod Laver Arena plenty to cheer, just 24 hours after they saw countrywoman and top seed Iga Swiatek bundled out.
Linette, 30, was almost overcome after reaching her maiden Grand Slam quarter-final in 1hr 57min.
"I don't really believe it," she said. "I still can't believe it, I don't know what happened.
"I'm speechless, really. I don't know what to say."
Belarusian Sabalenka never really looked like she would follow Swiatek, Garcia and the rest out of the first major of the year.
Along with third-seeded American Jessica Pegula, fifth-seed Sabalenka looks the woman to beat. Both are chasing a maiden major crown.
Sabalenka defeated dangerous 12th-seed Belinda Bencic 7-5, 6-2 to set up a last-eight clash against unseeded Donna Vekic, who beat 17-year-old Linda Fruhvirtova in three sets.
Hard-hitting Sabalenka sprinted into her first quarter-final at the Australian Open and said: "My whole life, it took me a little while to understand that negative emotion is not gonna help you on court.
"You have to just stay strong and believe no matter what and then do everything you can."
Garcia latest Australian Open casualty but Sabalenka unstoppable
Caroline Garcia Australian Open Aryna Sabalenka