Williams, seeking an Open-era, record-equalling 22nd Grand Slam title, was a set up and about to tackle a second set tie-break against France’s Kristina Mladenovic when a thunderstorm sent everyone scurrying for cover.
On the resumption, more than two and a half hours later, the top seed and defending champion needed five match points to finish the job going through 6-4, 7-6 (12/10) with Ukraine 18th seed Elina Svitolina next in her line of fire following her victory over former champion Ana Ivanovic.
“It was very difficult, she played really, really well. She forced me to bring out my best tennis,” the 34-year-old American said.
Thiem was already done and dusted ahead of the rain, downing German teenager Alexander Zverev 6-7 (4/7), 6-3, 6-3, 6-3 to reach the fourth round.
In what was a clash of two of the sport’s rising new stars, the Austrian dropped a tight first set, but then coasted away from his 19-year-old opponent.
He will next play Spain’s Marcel Granollers, who reached the fourth round without hitting a ball after nine-time champion Rafael Nadal withdrew due to a wrist injury.
If he wins that, it will be the first time that 22-year-old Thiem has reached the quarter-finals of a Grand Slam tournament.
“It’s been the third match in the last four weeks between us and it was really tough, but I knew that from the start already,” he said.
“It was my first match on one of the big stadiums (at Roland Garros) and it was packed. An unbelievable experience.”
Also through to the last 16 was 2014 losing finalist David Ferrer who comfortably saw off the challenge of fellow Spaniard Feliciano Lopez 6-4, 7-6 (8/6), 6-1
In the women’s tournament, there were early wins for eighth seed Timea Bacsinszky of Switzerland, Spain’s Carla Suarez Navarro and Kazakhstan’s Yulia Putintseva, who reached the last 16 of a Grand Slam tournament for the first time.
The long rain delay forced organisers to shuffle the court scheduling with notably top seed Novak Djokovic switched from the Philippe Chatrier centre court to the secondary Court Suzanne Langlen for his match against Britain’s Aljaz Bedene.
The Serb has already seen his chances of becoming just the eighth man to win all four Grand Slam titles hugely helped by Nadal’s shock withdrawal and his only previous match against Bedene was a straight sets win in the first round of last year’s Australian Open.
Both the Williams sisters were on the schedule ahead of a potential semi-final showdown with Venus also taking on a home hope in the shape of Alize Cornet.
Cornet made headlines in the last round when her vanquished opponent, Tatjana Maria of Germany, angrily accused her of faking injury to gain time to recover from a severe bout of cramps.
The bad feeling from that clash spilled over with Maria saying that her camp were considering taking legal action for a violation of the rules against the International Tennis Federation, the WTA or the French Open