Australian world number 108 Kokkinakis came through 3-6, 7-5, 6-3, 6-7 (4/7), 6-3 in four hours and 38 minutes against the 2015 champion.
"I feel better now that I've won," said Kokkinakis after securing victory on a fifth match point.
"What a match. I know Stan is getting older but in the first set, he was playing the best. I was nowhere, just hanging in."
Kokkinakis, who described Wawrinka, a three-time major winner, as a "legend" of the sport, last made the third round of a Slam in Paris eight years ago when he was just 19.
However, he has been plagued by physical and medical problems with a shoulder injury limiting him to just one singles match from November 2015 until May 2017.
He then missed the entire 2020 season due to mono before undergoing surgery on his sinuses in 2022.
"I've had a pretty crazy story. I came on the scene when I was pretty young, looked like I was set for a big future but I went missing for a few years," he said.
Kokkinakis will take on Russian 11th seed Karen Khachanov, who beat Moldova's Radu Albot, for a place in the last 16.
