Thiem, the runner up to Rafael Nadal in 2018 and 2019, saw off the 20-year-old 6-4, 6-4, 5-7 3-6, 6-4 after three hours and 32 minutes under the roof on Court Philippe Chatrier.
"It was an amazing match from both of us," said Thiem. "And such fighting qualities from him. I haven't seen such a good touch from a player in a long time," added the US Open champion. "All the credit to him and if he continues like this, he's going to be a huge player and he will give a lot of joy on this court."
For the first couple of hours, the form book was respected as Thiem kept the willing left-hander at bay with strong service games.
Wilting
But towards the end of the third set the Austrian seemed more vulnerable and from having an impregnable serve, he lost it twice to cough up the set.
Revitalised and roared on by the partisans on centre court, Gaston, the world number 239 who beat former champion Stan Wawrinka in the third round, led Thiem a merry dance in the fourth set with dazzling patterns and a barrage of drop shots.
Gaston, who was invited by the French Open organisers to play in the main draw, won that 6-3 and continued to hold his nerve during the decider.
Ultimately, the drop shots that had brought him back from the brink were his undoing. Thiem capitalised on the a few errors to make the break to lead 5-4.
But even when serving for the match, the Austrian had to stave off a break point before clinching the tie and progressing to his fifth successive French Open quarter-final.
"His drop shots are from another planet... I was sprinting like 400 times to the net" 😆@ThiemDomi impressed (if not a little tired) from some of those drop shots from Gaston...#RolandGarros pic.twitter.com/fIavNlIjwX
— Roland-Garros (@rolandgarros) October 4, 2020
Thiem will play Diego Schwartzmann from Argentina in the last eight after the 12th seed eased past the unseeded Italian Lorenzo Sonego 6-1, 6-3, 6-4.
Earlier in the men's last 16 in the bottom half of the draw, defending champion Rafael Nadal swept into a 14th French Open quarter-final with a straight sets win over 20-year-old Sebastian Korda from the United States.
The 34-year-old Spaniard will take on the 19-year-old unseeded Italian Jannik Sinner who upset the sixth seed Alexander Zverev in four sets to become the first man since Nadal to reach the last eight on his first appearance at Roland Garros.
