The match took almost three hours on the hottest day of the tournament so far, and the 37-year-old Williams now must recover quickly to next play defending champion Johanna Konta.
After dropping a grueling first set, Venus bounced back strongly to register a 5-7, 6-1, 6-2 victory over her Australian opponent in a thrilling last-16 clash that lasted for more than two hours.
Williams squandered an early lead, fell behind late and then overcame three match points to rally past Kiki Bertens 5-7, 6-3, 7-5 in the third round.
Konta made a roaring start by breaking in the first game of the match and she nearly repeated the feet in a marathon third lasting nearly 11 minutes only for Williams to recover.
When Williams slammed a forehand victor on the last point, she spun with a grin in celebration, her fist leading the way. "It didn't look good sometimes but that's why you keep playing to the last point and why I love this game". The Briton now awaits the victor of the match between Venus Williams, the seven-times grand slam champion, and Kiki Bertens, the world No 29 of the Netherlands.
Williams, ranked eighth in the world, will battle for a quarter-final berth against defending champion Johanna Konta, a 6-2, 6-1 victor over Belgian Elise Mertens. Konta broke the Williams straight away to set down an early marker in their first meeting since the semi-finals at Wimbledon last summer which resulted in home heartbreak for the 26-year old.
Konta got off to the flawless start, breaking Williams in the opening game.
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The Dutch, bidding for the fourth top 10 win after beating No.3 Kerber and No.9 Bacsinszky (2016 Roland Garros) and No.7 Vinci (2016 Nürnberg), got broken after a marathon game to 0-2 in the second set.
Latvian Ostapenko defeated Brazilian Beatriz Haddad Maia 6-2, 7-6 (7/2).
As Williams stormed to a 6-1 second set to set up a decider, Konta called out the trainer, seemingly struggling with her lower back.
Ostapenko twice served to avoid a third set.
"In the tie-break, I showed my great tennis, and I won the match", Ostapenko said.
American qualifier Danielle Collins will take on Olympic gold medal victor Monica Puig later in the day to determine who Williams' next opponent will be.