Second seed Osaka, the favourite for the gold medal following world number one Ashleigh Barty's first-round exit, beat the Swiss 6-3 6-2.
Osaka will face Marketa Vondrousova, who beat Mihaela Buzarnescu 6-1 6-2, for a quarter-final spot.
World number one Novak Djokovic beat Germany's Jan-Lennard Struff 6-4 6-3.
Djokovic is bidding for a first Olympic gold medal after winning bronze in Beijing 2008.
Great Britain's Jamie Murray and Neal Skupski were knocked out of the men's doubles in the second round, losing 6-3 6-4 to Japan's Kei Nishikori and Ben McLachlan.
Osaka, 23, took an eight-week break before the Games in order to protect her mental health.
She described making her comeback in the Olympics as being an "interesting" experience.
"The flag is next to my name no matter what tournament I play, but I feel like the scale of this is much bigger," she told American TV.
"It's something I've been waiting for for eight years."
There were also speedy victories for Garbine Muguruza and Anastasia Pavlyuchenkova.
Spain's two-time Grand Slam champion Muguruza saw off China's Wang Qiang 6-3 6-0 in 61 minutes.
French Open runner-up Pavlyuchenkova, representing the Russian Olympic Committee, took only 54 minutes to beat Germany's Anna-Lena Friedsam 6-1 6-1.
Czech former world number one Karolina Pliskova progressed with a 6-3 6-7 6-1 victory over Spain's Carla Suarez Navarro.
Germany's Alexander Zverev advanced in the men's singles, beating Colombia's Daniel Elahi Galan 6-2 6-2.