Stunned by Mladenovic in Dubai last month, Osaka had no such problems in California, claiming a 6-3 6-4 victory in the second round.
Wawrinka won a taut thriller against 29th-seeded Hungarian Marton Fucsovics 6-4, 6-7 (5/7), 7-5, finally sealing it on his fifth match point after three hours and 24 minutes when Fucsovics lofted a lob long.
Osaka will next face Australian Open semi-finalist Danielle Collins, who thrashed Kirsten Flipkens.
Eager to avoid a repeat, Osaka marched through the opening set in 38 minutes, but hit a speed bump as she was broken when serving for the match at 5-2 in the second.
Watch: Djokovic Prepares For Indian Wells With Practice Session With WawrinkaThe Serbian will be playing for the first time since winning his 15th Grand Slam title at the Australian Open on 27 January.
The Chinese player nearly took the first set in a tight tenth game, holding two set points on the former world No. 10's serve.
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But Sheriff Jay Jones said the list of missing people had shrunk from dozens to just seven or eight. He described the damage as "catastrophic", and said he expected the death toll to rise.
"I was feeling good".
"Don't remind me of the stats", Wawrinka quipped after gutting out a 6-4, 6-7 (5/7), 7-5 second-round victory over Marton Fucsovics. I didn't play my best tennis today, but on certain points where I maybe had a little sniff or a little chance to get into the match ... he came up with a really good shot to break me in the first set. It was just her eighth match of the year.
"Having to play down the whole match and to be able to get that", Williams said of her emotion. "I know most of my team members at the time were against me playing in Indian Wells and Miami because it was too early after the surgery", Djokovic said on Thursday during his pre-tournament press conference. "I did go through doubtful moments and questioning everything and experiencing a surgery for the first time in my life was something that I've never experienced before".
Earlier, Roger Federer began his campaign with a comprehensive victory over Peter Gojowczyk to ease into the round of 32.
Two-time Wimbledon champion Kvitova, who was runner-up to Osaka in Melbourne, fired 10 aces, but also had 10 double faults, and her 56 unforced errors were more than double the American's 25.
Third seed Alexander Zverev is also through after opening opponent Martin Kližan of Slovakia retired when trailing the German 6-2, 2-0.