Thiem, a semi-finalist at Roland Garros for the past two years, has won eight of his 10 ATP titles on clay and dominated on his favoured surface.
Germany´s Alexander Zverev celebrates after victory over Bosnia-Herzegovina´s Damir Dzumhur at the end of their men´s singles third round match, on day six of The Roland Garros 2018 French Open in Paris on June 1, 2018.
Dzumhur would go on to lose the match in a five-set thriller, with second seed Zverev saving match point to go through to the fourth round.
In women's play, 13th-seeded Madison Keys reached the quarterfinals for the first time at Roland Garros after beating No. 31 Mihaela Buzarnescu of Romania 6-1, 6-4 on Chatrier.
The 28-year-old, seeded 19 in his first Grand Slam since Wimbledon last year after struggling with a wrist injury, saw off Simon 6-3, 6-1, 6-3.
But he has never reached a Grand Slam quarter-final, although the longer best-of-five format has worked twice in his favour now at Roland Garros.
As you can see in the video above, the young ball boy came off second-best. "This definitely paid off, the hours in the gym every day ..."More news: Good news for Chelsea as Lewandowski is set to leave Bayern Munich
More news: School shooting video game taken down after widespread backlash
More news: Wait, Does Roseanne Barr Really Think Michelle Obama Got Her Show Canceled?
He will face French No.15 seed Lucas Pouille or Karen Khachanov in the fourth round. "I'll get myself ready", Zverev said, smiling.
The 34-year-old Verdasco will next face either his compatriot Roberto Bautista Agut or former champion Novak Djokovic for a quarter-final spot.
Dzumhur served for the win in the fourth set but never held a match point, with Zverev fighting back in the tiebreak. Djokovic advanced to his 40th Grand Slam quarter-final.
Ottawa's Gabriela Dabrowski and Croatia's Mate Pavic cruised into the quarter-finals of the mixed doubles tournament with a 6-2, 6-3 win over France's Eduoard Roger-Vasselin and Andrea Sestini Hlavackova of the Czech Republic on Sunday.
Stephens next plays either Australian Open champion Caroline Wozniacki - seeded second - or 14th-seeded Russian Daria Kasatkina.
Keys, the US Open runner-up to Stephens a year ago, will face Kazakhstan's Yulia Putintseva for a place in the semi-finals, after the world number 98 saw off Barbora Strycova 6-4, 6-3. But Dzumhur felt it was within his reach and also ran out to catch it.