Mladenovic's 94-minute victory booked her a blockbuster clash with Australian Open champion Osaka after she upstaged the Japanese star at the Dubai Tennis Championships in February.
"I've never been a defending champion before, that's new and I was really nervous", said Osaka, who booked a third-round meeting with American Danielle Collins, a 6-4, 6-1 victor over Kirsten Flipkens.
Williams lost the first set 6-3 to two-time major champion Garbine Muguruza and was trailing 1-0 in the second set when she called it quits.
A break to begin the second set put her in complete control before another gritty hold for 4-2, although there would be a brief stumble.
Djokovic was playing his first match since winning the Australian Open in January and conceded an early break to his 25-year-old American opponent, ranked 128th. And I think I created some opportunities.
The number five seed dropped the first set in a tie-break but made light work of the remainder of the match, prevailing 6-7 (4), 6-1, 6-1.
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After she went missing her family put up a Facebook page titled "Help find Preethi Reddy ". Inside the restaurant, footage shows Dr Reddy standing alone at the counter.
Djokovic beat Bjorn Fratangelo 7-6 (5), 6-2 in pursuit of a record sixth title in the desert. Summing up how she felt after the match, Kvitova told reporters: "I want to swear, but I will not".
When I see someone at the net, I just start thinking percentages, like, you know, 30% chance they are going to go this way, so might as well go with the higher chance - or she went this way last time, so she might go this way the other time. "If I maybe would have served a little bit better, I think things would have gone faster".
The 25th-seeded American beat Belgian Kirsten Flipkens, 6-4, 6-1. "I was so frustrated with myself from the half of the second set to the end, which shouldn't happen to me", she further explained.
Donaldson's reply was pretty typical for a World No.192 who's just about to play a 17-time grand slam champion.
Barthel's compatriot Angelique Kerber was almost flawless en route to a 6-0 6-2 dismantling of Yulia Putintseva, winning 93 percent of her first service points under sunny skies in the Southern California desert.
Russian 14th seed Daniil Medvedev is also through after he dismissed American Mackenzie McDonald 7-5 6-0 and he will face Filip Krajinovic next after the Serb conquered 20th-seeded Belgian David Goffin 6-3 6-3.