The world number 62, who has dropped from a career-high of 17th a year ago, fought back from a set down to win 5-7, 6-3, 6-2, 6-4 and reach the first Grand Slam quarter-final of his career.
British fifth seed Draper was a heavy favourite after a strong season so far in which he has won the Indian Wells title and reached the Madrid Open final.
But Bublik produced an inspired performance to set up a last-eight tie at Roland Garros against either world number one Jannik Sinner or Russian Andrey Rublev.
He sealed the win in a dramatic final game, in which Draper missed five break points, on his second match point.
He won 79 per cent of his first-serve points and converted on five of six break points to achieve his first win over Draper in three meetings.
Bublik is the lowest-ranked man to earn two top-10 wins at Roland Garros since No. 100 Andrei Medvedev beat Pete Sampras and Gustavo Kuerten to reach the 1999 final.
"I left it all out there. 100 per cent one of the best days of my life and one of the best matches I have ever played in my life. That was as simple as that," said the 27-year-old.
"You know, sometimes in life there's only one chance and I had the feeling that that was mine and I couldn't let it slip. Standing here is the best moment of my life, period.
"You know I'm standing here like I won the thing, but at the end what can I say... Thank you guys. I can't cry here, come on, stop. Let me be in peace, I still have a match to go. I'm a professional tennis player, I've got to get ready."
Not known for his skills on red clay, all of Bublik's four tour-level titles and 11 finals have come on either hard or grass courts. He had reached the fourth round once before at a major, at Wimbledon in 2023.
"Maybe this is the first year I haven't complained about playing on clay, as I don't have a lot of options because I was dropping a lot in the rankings. Yeah, I guess that's the key," he continued.
This season, he has also reached the fourth round at the Madrid Masters and won the Turin second-tier Challenger Tour event on clay.
"I thought he played incredibly well. He didn't allow me to play my tennis," said 23-year-old Draper.
"A couple of dips here and there, and he took advantage. It hurts a lot. I had an opportunity today, and I missed my opportunity, for sure."
