The Ukrainian world number four breezed past her compatriot 6-2, 6-2 in just 59 minutes on a searing day on Rod Laver Arena.
"It's very special for me to get past the third round," said Svitolina, who will be in the last 16 for the first time in Melbourne where she will face another qualifier, Denisa Allertova.
"It's sad I was playing another Ukrainian girl."
Kostyuk had been labelled the "future of tennis" by some observers after becoming the youngest Australian Open second- round winner since "Swiss Miss" Martina Hingis in 1996.
"She's definitely got a bright future," said Svitolina. "We'll be hearing a lot more about her.