Thompson saved three match points before pulling off an epic 4-6 7-6 (7-1) 7-6 (9-7) victory over American Aleksandar Kovacevic in three hours 34 minutes - the longest singles encounter in tournament history.
The Australian Davis Cup star said he didn't know how he escaped defeat.
"I was really struggling on serve but somehow found a way and I guess that's the beauty of clay," Thompson said.
The 29-year-old's rewards are significant.
In addition to ensuring a fresh career-high ranking of at least No.31 in the world and a last-eight showdown with defending champion Frances Tiafoe, Thompson is now on track to be seeded at next month's French Open.
The Aussie No.2 must remain inside the top 32 for another five weeks to earn the all-important early-round protection in Paris.
At this rate, that shouldn't be a problem.
A winner over the great Rafael Nadal in Brisbane in January, Thompson is flying.
He landed his maiden ATP title in Los Cabos on hard courts in February after taking out world No.6 Alexander Zverev in the semi-finals and three-time grand slam runner-up Casper Ruud in the title match.
Jordan Thompson raises the trophy after winning in Mexico. (EPA PHOTO)
Now Thompson, a one-time grass-court conqueror of then-world No.1 Andy Murray at Queen's, is once again showcasing his clay-court prowess.
He was Australia's last man standing at Roland Garros in 2019, after reaching the third round, and would be guaranteed to avoid a higher-ranked rival until at least the last 32 if he manages to jag a seeding this year.
His biggest concern is juggling doubles duties with compatriot Max Purcell and needing to return to the court in 18 hours to face Tiafoe.
"Pretty fair schedule? Not too happy about it so I won't say much more," Thompson said.
At least he has experience.
As well as his midnight thriller with Nadal, Thompson finished at two in the morning in Mexico to complete his double act in February.
Tiafoe stands in his way next after beating Thompson's fellow Sydneysider James Duckworth 6-2 6-7 (4-7) 7-5 earlier on Thursday night in the US.
The third-seeded American fired down 15 aces and didn't face a break point against Duckworth, who did well to take the match deep into a third set.
"Every win is big right now, to just kind of build it," Tiafoe said.
"Tonight was huge. I wish I won in straights, but I definitely played pretty well.
"He can serve really well. I had some chances, some break points, but good serves from him and he really made it tough for me."
Eighth-seeded Purcell also lost to an American, 6-2 7-5 to wildcard Michael Mmoh.
Mmoh will face fourth-seeded Argentine Tomas Martin Etcheverry, a 6-4 6-4 winner over Colombian Elahi Galan, for a place in the semi-finals.
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