For the second match in a row Osaka had to battle back from a shocking start, this time losing the first four games against former world number one Victoria Azarenka before winning a titanic tussle 4-6 7-5 6-3 in just under three hours.
The Japanese top seed struck 52 winners as she stretched her winning run in the Grand Slams to 16 matches.
"I have this mindset that I feel like I can win if it gets down to the wire, like, if I have to break a person, I feel like I have the ability to do that," Osaka, who plays Katerina Siniakova of the Czech Republic next, said.
Djokovic, bidding to hold all four Grand Slams simultaneously for the second time in his career, had no such dramas as he followed Osaka on to Court Suzanne Lenglen.
The Serb was in cruise control as he beat Swiss lucky loser Henri Laaksonen, the world number 104, 6-1 6-4 6-3 to set up a third-round clash with Italian qualifier Salvatore Caruso.
"He has nothing to lose. I'm sure he's going to be very motivated to play his best," Djokovic, who won the title to complete his first "Nole Slam" in 2016, told reporters.
Serena Williams moved smoothly through to the third round with a 6-3 6-2 defeat of Japanese qualifier Kurumi Nara, setting up an all-American tussle with Sofia Kenin who benefited from the withdrawal of 22nd seed Bianca Andreescu through injury.
Williams is trying again to match Margaret Court's 24 Grand Slam titles having fallen twice in finals last year on her comeback from maternity leave.
Defending champion and third seed Simona Halep did not have things all her own away against Poland's Magda Linette but came through a tough battle 6-4 5-7 6-3.
Dominic Thiem, beaten in last year's final by Rafael Nadal, also had some trouble against Alexander Bublik.
He had to save two set points in the fourth set to avoid a decider and was relieved to win 6-3 6-7(6) 6-3 7-5. He will play unseeded Pablo Cuevas next.
Fifth seed Alexander Zverev, ninth seed Fabio Fognini and 13th seed Borna Coric all progressed with ease although Juan Martin del Potro, the eighth seed, needed five sets to subdue Japan's Yoshihito Nishioka.
Several Frenchmen survive in the draw, including Gael Monfils who will face a second successive all French clash in the third round when he plays wildcard Antoine Hoang.
Hoang enjoyed a memorable win on the Court One "Bullring" against Spain's experienced 23rd seed Fernando Verdasco.
No French women have reached the third round for the first time since 1986 after Caroline Garcia lost 1-6 6-4 6-4 to Russian qualifier Anna Blinkova.
(Reporting by Martyn Herman; Editing by Toby Davis)
