Ajla Tomljanovic has caused one of the big early shocks at the French Open, knocking out world number five Anett Kontaveit on a red-letter day for Australia's top two tennis women.

After Daria Saville continued her stirring comeback to international tennis, breezing through her first-round match against Valentini Grammatikopoulou earlier in the day, she declared she was now after the Australian number one spot that she held briefly in 2017.

But Tomljanovic, the current holder of that accolade, showed why she's still the tops in the post-Ash Barty era, powering past fifth-seed Estonian Kontaveit to secure one of the best wins in her career.

In a red, white, and blue outfit that had been designed to get the home crowd on her side, Tomljanovic, with her right thigh still heavily strapped after having to pull out of last week's Morocco Open, showed no ill-effects as she largely bossed the contest to win 7-6 (7-5) 7-5.

She'll next play Varvara Gracheva, and will recognise the danger after the Russian knocked out another Australian, Astra Sharma, on Sunday.

At the start of a damp day, "Dasha" Saville had wasted no time in dashing into the second round with a 6-1 6-2 win over the outclassed Grammatikopoulou.

It took the 28-year-old less than an hour and a quarter to beat the Greek qualifier — and the Paris rain — with the win ensuring she will leapfrog third-ranked Maddison Inglis and current Australian number two Sharma when the new rankings are released.

"I don't look at rankings so don't tell me!" laughed Saville, who's made extraordinary progress since returning from 10 months out following surgery to deal with a chronic Achilles injury.

But at the same time, she couldn't help admitting: "One of my goals is to be the number one Aussie.

"It's a privilege to be nervous. It shows I really care. And even though I was nervous today, I was still able to perform well."

The pair's progression followed a dramatic day in the women's circuit, with defending champion Barbora Krejcikova and four-time grand slam winner Naomi Osaka both making early exits from the Open.

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Krejcikova becomes just the third reigning champion to be knocked out in the opening round of Roland Garros after losing to 19-year-old, 97th-ranked Frenchwoman Diane Parry.

"It's a dream for me," Parry said after her career-best win.

Osaka, meanwhile, said she's reconsidering her tilt at Wimbledon following her elimination by 27th seed Amanda Anisimova, the 20-year-old American who defeated Osaka in the third round of the Australian Open earlier this year.

Angelique Kerber almost followed the two high-profile exits, but survived two match points against Poland's Magdalena Frech in a two-and-a-half-hour epic to progress 2-6 6-3 7-5.

The 34-year-old Kerber, who has never gone beyond the quarter-finals on Paris clay, was 3-5 down in the third set. She faced two match points at 4-5 down but broke back, held, then broke again to win.

Other winners in the women's tournament included world number one Iga Swiatek, as well as former grand slam champions Petra Kvitova, Victoria Azarenka, Emma Raducanu, and Bianca Andreescu.

In Australia's other match of the day, Jordan Thompson battled hard but simply had no chance against 13-time champion Rafael Nadal, who steam-rolled past the challenger 6-2 6-2 6-2 on Court Philippe Chatrier.

AAP