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: Pedro Sánchez sworn in as Spain’s prime minister
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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.